This is a traditional item, with no records of its authorship. Kauthuvams are performed in the temples when the deities, kauthuva murthis, are taken out on procession as part of the temple ritual. In kauthuvams, unique rhythmic phrases depicting the deities are sung and the rhythmic phrases are recited, after which fast nritta passages follow.

Translation of Kali kauthuvam:

This story is about Kali dancing in Aanakadu Tholaada
Kali's devotee sage Patanjali and Vyagraprada were praying to catch a glimpse of Shiva and Shakti dancing together.
Nandi was the nathuvanar, setting the beats. Narada was singing.
Sun-god and Moon-god were there, sharing this joint undertaking
There were 108 different beats.
To the 108 beats the divine feet of Shiva danced.
Shiva danced, the sage Thandu danced and the gods celebrated this cosmic dance in Thillai (in Chidambaram).

Kali kauthuvam is about the black goddess, symbol of dissolution and destruction. she destroys ignorance, maintains the world order, and blesses and frees those who strive for the knowledge of god. She is the destroyer of all suffering and all evil. she has fang-like teeth, and her lolling tongue dripping with blood hangs from her mouth. around her neck is a necklace of skulls. she is usually shown with four arms, two of which hold severed heads while the other two hold a dagger and a sword. a strangling noose also features in some of the images. though her hands are blood stained, one is often raised in a gesture of protection or assurance in the midst of destruction.
to see shiva’s blissful dance with Shakthi that energizes the worlds, two ardent sages vyagrapada and patanjali were worshipping Shiva in Thillai near Chidambaram, accompanied by other sages, and the sage Narada, the Sun-god, the Moon-god, and the cosmic snake Shesha.
at the same time Bhadrakali, the fierce form of Shakthi asked for a competition with shiva in dance. shiva and mahakali came to the thillai, where the sages were performing austerities to see the divine dance. the lord, who keeps dancing creating, protecting, destroying, hiding and blessing, won the play by acting the urdva tandava, raising his feet to his head.
Kali kauthuvam mentions it that Shiva’s divine feet were dancing to 108 talas.
In Sama gana, from whicn carnatic music is derived, the 108 talas, or beats, represent different divine aspects.

Translation:

Your compassion is greater than the ocean.
you have a beautiful face with three eyes, and you have matted hair.
You have Parvathy as your other half,
Sadasivam Rudram,
oh Lord of Chidambaram, you are in every form
I worship you whole-heartedly.
Oh Great God Shiva!
You are 1000 times more beautiful than Manmadan, god of love.
You protect millions of people,
you bless the sages.
The supreme God, you are the god of grace.
You like that chants of Samaganam.
Oh Supreme Ruler, you bear the river Ganga in your hair.
Oh Divine Master of the lower Nature, You are the most pure one

The item is about Lord Shiva who is described as the one who likes the chants of Sama Veda. The traditional Bharatanatyam items are always accompanied by the Carnatic music, which is based on the musical system of Sama Veda, which has mystic dimensions.
Shiva is inseparable from Shakti - Parvati the daughter of Himavaan - Haimavati. There is no Shiva without Shakti and no Shakti without Shiva, the two are one - or the absolute state of being - consciousness and bliss.
Shiva is called Pashupathi, the Lord of the lower nature. Pashu is often misinterpreted as “cows”. The difference between man and animals lies in the number of legs: man is a Dvipaadha Pashuvu (two legged animal) and animals are Chatushpaadha Pashuvu (four legged animal). Having been born as human being, you should strive to become Pashupathi (the Master of your lower, animal nature). Do not degenerate to the level of Pashu (animal).
Rudra-Shiva is Mahakala (the Great Time-Lord) and Mrtyunjaya (Death Conquering).  Rudra is the Red God (Rohita), intimately associated with life’s blood/fire; and Shiva, as the agent of change (Rajas), has power over life and death. Shiva is eternal, and driven to manifestation only through the Tapasya of devotees, to whom he grants secret knowledge and magical weapons.
He is given a range of titles which include Maha-deva (great god), Nata-raja (lord of the Dance).  Shiva has over 100 names, including Shankar and Shambhu. Shiva is the destroyer of our illusion and ignorance that stands in the way of our perfect union and enlightenment.
Ganga, symbolically represented on the head of the Lord by a female (Mother Ganga) with a jet of water emanating from her mouth and falling on the ground, signifies that the Lord destroys sin, removes ignorance, and bestows knowledge, purity and peace on the devotees.
The three matted locks on the head of the Lord convey the idea that integration of the physical, mental and spiritual energies is the ideal of yoga.
The Lord's right hand is shown in a boon- bestowing and blessing pose, Varda Mudra. Trident (Trisula) symbolizes His three fundamental powers (shakti) of will (iccha), action (kriya) and knowledge (jnana). The trident also symbolizes the Lord's power to destroy evil and ignorance.
Lord Shiva, sitting on or wearing a tiger skin, illustrates the idea that He is the source of the creative energy that remains in potential form during the dissolution state of the universe. Of His own Divine Will, the Lord activates the potential form of the creative energy to project the universe in endless cycles.

Translation:

Uma Maheshwari,
you are the Heroine of the Universe.
Bless us, oh wife of Shankara, the kind one,
You move seated on your throne of Lion. Oh, the embodiment of divine Love!
Oh the one who is always yearning for Siva, your dance is fascinating!
Your face is like a red flower.
Oh, the supreme triune beauty,
you killed the demon Pandasuran,
and you blessed us.
Oh beautiful Ambigai (Mother),
You were standing with your foot lifted in the middle of a raging fire, doing your penance.
You are the beginning and the end of life.
You bless us with 1000 eyes.
You are seated in the center of the Sun chakra.
You give us the wonderful pleasure, Akilaandeshwari (the Empress of the World).
Rajarajeshwari, you dance to ragam, talam and bhavam.
Holding a sugarcane in your hand,
you bring good fortune to your devotees.
Oh, the mother of the universe, You are wearing a fascinating emerald crown.
You are the sister of the mischievous Vishnu.Bless us whole-heartedly, oh the One Holding the Trident.
Without you, life does not exist.
You elevate us. You bring goodness to people.
Protect us, oh the Compassionate One.
Why the delay, oh Mother?
Muruga on the colourful peacock,
and elephant-faced Ganesh worship you, Mother.
Ruler of Madurai, oh Meenakshi.
You live in the heart of the poet.

The varnam describes the Great Goddess who abides in all beings.
She is the goddess of the supreme knowledge, and brings to us her vision for all kinds and widenesses of truth, her rectitude of the spiritual will, the calm and passion of her supramental largeness, her felicity of illumination. She is Mahakali, goddess of the supreme strength . With her are all mights and spiritual force and severest austerity of tapas and swiftness to the battle and the victory and the laughter, the attahasya, that makes light of defeat and death and the powers of the ignorance
the Great Goddess, Tripura Sundari, is also called Shodashi, Lalita and Rajarajeshvari, is one of the group of ten goddesses of Hindu Belief, and these goddesses are collectively called mahavidyas. The other nine mahavidyas are Kali, Tara, Bhuvaneshvari, Bhairavi, Chhinnamasta, Dhumavati, Bagalamukhi, Matangi, and Kamalatmika. The Tripura Sundari Ashtakam by Adi Shankaracharya describes her as a Mother, her nourishing breasts akin to mountain peaks that give birth to rivers. She is described wearing a blue dress with red spots, holding a pot of honey, her eyes quivering with intoxication.
My (“maam”) obeisances (“paahi”) to the goddess Raja Rajeshwari, who is the kind one (“krupakari”) and the wife of Shankara (“shankari”). She is the one who bestows (“dehi”) unsurpassed happiness (“ehi sukham”). She is the one who rides (“vahini”) on a lion (“simha”). She is a river of kindness, and the enchanting one. She is the one who destroyed (“khandani”) the demons: Bhanda, Chanda, Munda and Mahisha. She is the entertaining one (“Ranjani”) and the spotless one (“Niranjani”). She is the one who takes care (“poshani”) of the learned “guhadaasa” (the composer). She has sweet (“su”) voice and speech (“bhashini”). She is the one who defeats (“bheeshani”) enemies (“ripu”)  and grants (“bhushani”) boons (“vara”).
The Devi Bhagavata describes the manifestation of the Divine Mother as Lalita in response to the prayer of the creator, Brahma to protect the celestials from the atrocities of Pandasura. This demon rose from the ashes of Manmatha when Siva burnt him to ashes for distracting Him while meditating. Lalitha Parameshwari after coming out from the 'Chitangi Kunda' was supposed to be Sworn (Rajya Patabishekam) the thrown of the Bhuvana in order to kill the demon Pandasura, that is, Lalitha Parameshwari became the queen of the Bhuva and was called 'Bhuvaneswari' and after the coronation was named as 'RajaRajeswari'.
To appreciate the female power (Shakthi) of Uma Maheshwari, the consort of Lord Parameshwara, one need to rewind a few centuries and understand the time of Mahishasura the demon, who was oppressing the three lokas. Mahishasura had become invincible. His attrocities on the other Gods and humans were unbearable and Indradi Devas approached both Mahavishnu and Parameshwara, under the leadership of Brahma,  for deliverance. The anger of Parameshwara and Mahavishnu at the impudence of  Mahishasura was so great, a new luminous energy emerged from their personalities and all the other lesser Gods contributed their might to add more force to this emerging personality.  From this store of energy a beautiful goddess was fashioned and she was named Katyayani, because she claimed parentage from the sage Katyayana. Shiva's energy formed her face, Agni donated  to her, the eyes, Yama gave her beautiful tresses, Mahavishnu true to his name gave her eighteen hands, Indra her feminine waist, Varuna provided her legs to walk, Brahma her feet, and Surya her toes.  The Yakshas for their part gave her a nose, the Sadhyas her eyebrows and the Maruts her ears.
She had an array of weapons also contributed by the other Gods. Shiva's Trishulam, Vishnu's Chakram, Varuna's Shanku, Agni's  spear-the Shakthi,  and Vayu's bow (chapa).  Surya gave his quiver (tuna) and arrows (shara).  Indra not to be left out gave his Vajra (Thunderbolt).  Yama the God of Death, gave his Danda (rod) and Kubera his Mace (gada).  Vishwakarma the celestial architect, gave her a new and powerful axe (kuthara).  Being a woman she was adorned with  fine jewellery and the great Himalayas gave her a huge roaring lion as her vahana (mount).
With this much of preparations, she descended on  the Vindhya Mountains and when the demon Mahishasura confronted her after a great and relentless battle she slew him and came to be called as Mahishasuramardhini, the slayer of Mahishasura.

Translation:

Oh, words cannot describe
the beauty of Shiva’s dance!
Thousands of sages worship you.
There is no place on this earth where you have not stepped.
The beauty of your perpetual dance
is inexplicable even with 1000 languages.
Your eyebrows bent bow-like dance gracefully.
Impeccably bringing out the navarasas, Shakthi also dances by moving her slender hips gracefully.
Many kind of beings dance to the rhythm of the music.
The crescent moon on the top of your head dances too. To crown it all your matted hair dances.
The entire river Ganga dances too.
Well-versed in the Vedas, the sages sing very well.
The tiger dances, the earth dances,and all the devotees dance too.
The Snake of the Universe dances too. The entire world is joyfully dancing, the beauty of which is beyond words.

This padam is about Lord Shiva, whose main attributes are the trident that represents the three gunas and the snakes that show he is beyond the power of death and poison and also stand for the Kundalini energy. The vehicle of Shiva is the white bull called Nandi (the joyful). He is often seated on a tiger skin or wears a tiger skin, with the tiger representing the mind.
Shiva Nataraj's dance represents both the destruction and the creation of the universe and reveals the cycles of death, birth and rebirth. His Dance of Bliss is for the welfare of the world. In the pose of Nataraj, the King of Dance is dancing Ananda Tandava and giving darshan to his beloved devotees within the "Hall of Consciousness" (ChitSabha). One hand is stretched across his chest and points towards the uplifted foot, indicating the release from earthly bondage of the devotee. The fire represents the final destruction of creation, but the dance of the Nataraj is also an act of creation, which arouses dormant energies and scatters the ashes of the universe in a pattern that will be the design of the ensuing creation.
Shiva is often depicted with a third eye with which he burned Desire (Kama) to ashes. Shiva bears on his head the crescent of the moon. Shiva's distinctive hair style is noted as "The One with matted hair, from which The Ganga river flows”. He is often shown seated upon a tiger skin.
Shiva's particular weapon is the trident. Shiva holds a small drum shaped like an hourglass is known as a "damaru". Mount Kailasa in the Himalayas is his traditional abode. Mount Kailasa is conceived as resembling a linga, representing the center of the universe.

Translation:

Narrated through the art of dance and musical theatre,
this enthralling story is about Rama and nine sentiments.
Like a mountain of distress the demoness Thaadakai came.
Her uproar shook the forest.
This woman demon that brought the misery was slain by Rama.
Thus arose the sentiment of joy.
Armed with a bow, while walking behind a sage,
Lord Rama’s holy footstep
transformed a stone back into a woman, freeing her from her curse.
Thus arose the sentiment of amazement.
In the maidens’ courtyard
the virgin Sita was looked upon by Rama
their eyes met and spoke a lot.
There arose the sentiment of love.
The noble Rama shot an arrow
While chasing the demon Maricha who had assumed the form of a mysterious golden deer.
The deer convulsed and fainted.
There arose the emotion of fear.
The wicked Ravana was fascinated by Sita and had cunningly abducted her.
Rama’s face, which was sweeter than a mother’s face,
became filled with the emotion of anger.
At the battlefield, Lakshman was bewitched by a serpent’s spell.
In agony, Rama with his arms cradled his brother.
There arose the emotion of sorrow.
Seeing that Ravana lost his weapons,
in front of the ten-faced Ravana, the chivalrous Rama came forward and said,
“Go home and come back with your arms”.
It demonstrated Rama’s knighthood.
After the triumph of the victory,
the devout Vibishana was crowned king of Lanka.
Blessed by Rama’s loving eyes,
the entire world witnessed the sentiment of grace.
When all was over,
everybody was happy.
The blissful Rama was crowned at Ayodhya on his throne.
Hosts of people witnessed Rama’s sentiment of serenity.

Nava means "Nine" and Rasa is often translated as "mood", "emotion", "expression" or "sentiment". Even though the Natya Shastra mentions only 8 rasas, the later texts refer to the nine basic expressions. Navarasa nilaye is based on Ramayana, and describes Rama in 9 episodes.
The first episode portrays how Rama killed the demoness Thaadakai. At Sage Viswamitra's insistence, Rama and Lakshmana accompanied him to the forest and there stood guard and helped him in successful completion of his yagas, slaying Thadakai and all the demons who dared to interfere. Appearing out of between two trees, Rama saw a horrible sight. It was the hideous form of the demoness, Tataka. She was enormous. Around her neck was a human skull. She had sharp claws on her hands.Taking careful aim, Rama fired his arrow which "pierced the chest of Thadakai and in its flight and speed passed through a mountain, a tree and finally even the very earth.  It is said Lord Rama had erected a temple for Siva near Bhoothapandy to get redemption from `strihatya' (slaying of woman).
The second episode is about Ahalya, who was wife of sage Gautama,  and who was turned into a stone. Vishvamitra while continuing the narration of the legend of Ahalya asks Rama to enter her hermitage where Ahalya is living unseen by anybody. When once Rama steps into that hermitage she manifests herself from her accursed invisible state. She emerges with her divine form and accords guestship to Rama and Lakshmana. Sage Gautama also arrives at this juncture to accept his wife Ahalya back.
In the third episode the meeting of Rama and Sita in the maiden’s courtyard is depicted as a meeting of their eyes. It was through their eyes that they spoke a lot first, and it was that exchange of looks that brought out the rasa of love.
The fourth episode speaks about rakshasa Maricha. Sita asks Rama to fetch the magical golden deer for her. The golden deer is in fact the demon Maricha, who must distract Rama and Lakshmana away from the hut so as to allow Ravana to kidnap Sita. The golden lustre of the deer lures Sita so much that she wishes to have the deer for herself. Sita insists on having it. Rama asks Lakshmana to take care of Sita while he goes for hunting the deer. While chasing the deer, Rama concludes from its behaviour that it is indeed unnatural and evil. Therefore he decides to kill it rather than capture it alive for Sita. After a long chase, he shoots it down with an arrow; but in its death-throes the deer cries out to Sita and Lakshmana for help, mimicking Rama's voice. Sita falls prey to the ruse and asks Lakshmana to go on a search for Rama. When Lakshmana insists that no one can harm Rama, Sita, still very much worried, implores and then orders Lakshmana to go. Although Lakshmana knows that Rama is invincible and beyond any danger, Sita panics and frantically orders Lakshmana to go to Rama's aid immediately. Unable to disobey the frightened and panicky Sita, and genuinely beginning to fear for Rama's safety, Lakshmana goes out to search for him. He however uses his mystical power to draw the Lakshmana Rekha or Lakshmana's Limit, a perimeter line around the hut that Sita must not cross, and no other being save Lakshmana or Rama may enter uninvited. If any intruder enters, it will be instantly killed.When Lakshmana leaves very reluctantly to look for Rama, Ravana succeeds in kidnapping Sita.
 As soon as Lakshmana disappeared from view Ravana transformed himself into a brahmin and approached the hermitage, taking care to stay well away from Lakshmana's line. Sita was in a dilemma. She cursed the golden deer and she cursed herself for sending first her husband and then her brother-in-law away. The brahmin was turning away. "Please wait!" she yelled. She went inside and returned with a pitcher of water and a bowl of fruits. She hurried to the line Lakshmana had drawn and stopped there. Sita, out of compulsion of religious duty and compassion for a poor brahmin, who is actually the disguised Ravana, crosses the line to give him alms. . In a trice Ravana leapt to her and held her in a vice-like grip. He reverted to his terrifying form and dragged a helpless Sita to his flying chariot. He was back in Lanka before the brothers returned to the hermitage and found Sita missing
The next episode portrays how during the battle to rescue Sita, Lakshmana is grievously injured by a mystical weapon fired by Indrajit, rendering him unconscious and approaching death. Rama and the others are shocked and in sorrow, when the army's physician explains that only with the special herb sanjivani that grows on the Dronagiri mountain in the Himalayan ranges, can Lakshmana be revived and brought back to life. Hanuman, the mighty vanara follower of Rama flies all the way across the length of Bharat to reach the mountain in the Himalayan ranges. Unable to find the magical herb, and knowing there was no time to waste, Hanuman uses his incredible might to lift the entire mountain and bring it to Lanka, the scene of the battle. The magical herb is retrieved and Lakshmana is revived.
The following episode describes Rama on the battlefield seeing an exhausted and disarmed Ravana dripping with blood and gore. Rama, known as the Karunasagara (ocean of compassion) saw it fit to tell him, “Go home today, return tomorrow”.
Vibishana, Ravana’s brother, was of a noble character and advised Ravana, who kidnapped and abducted Sita, to return her to Rama. When Ravana did not listen to his advice, Vibishana went over to Rama 's side and joined him. Later, when Rama defeated Ravana, Rama coronated Vibishana as the king of Lanka.
The end of the war coincides with the end of Rama's exile. Rama returns to a joyous Ayodhya. His mother, brothers and the people joyously welcome him. Kaikeyi is repentant of her deeds, and Rama forgives her. The next day, Rama is invested as the King of Ayodhya, and Emperor of the World. There was perfect justice and freedom, peace and prosperity.