I have found very exact and compact description of Varnam as musical composition in this blog: http://arvindsdad.blogspot.com/search/label/Varnam. There are also many links to music files.
Varnam is a form of song in the Carnatic music repertoire. A varnam is a relatively long piece and can range from 30 minutes to up to an hour. It is usually set to Aadi or Ata tala. It is the center piece in a recital of music or dance. The lyrics are simple and consist mostly of long syllables and swara phrases of various lengths which bring out the essential features of the raga. It has two types: Taana varnam and Pada varnam.
Varnams are considered as vocal exercises in a particular raga. The patterns in a varnam are characteristic patterns of a particular raga. Varnams are considered as the complex of the vocal exercises in Carnatic Music. They are designed to help develop voice culture and proper control of rhythm. Indeed, varnams are often practiced in double and triple speeds and proper rhythmic control (tala) must be kept.
Types of varnams
Named for its thanam-like rhythmic qualities, tana varnams only have lyrics for the pallavi, anupallavi and charanam.
With rhythmic elements like a padam, pada varnams are generally sung to accompany South Indian classical dance, including Bharatanatyam. Unlike the tana varnam which only has lyrics for the pallavi, anupallavi and charanam and swarams for the rest of the sections a pada varnam also have lyrics that correspond to the muktayi and chittai swaras of the varnam, so generally, pada varnams contain more lyrical content than a tana varnam. The swaras in this type of varnam are suitable for intricate footwork. Padajathi varnams are simply pada varnams that also contain jatis in it, making it again more suitable for South Indian classical dance.
Contents of a varnam
The varnam is subdivided into several sections:
• Pallavi: The first section of the Varnam, sung with lyrics or sahithyam.
• Anupallavi: A sort of recapitulation, sung with lyrics or sahithyam also.
• Mukthaayi Swaram: Sung completely with syllables—or swaras -- (like sa ri ga ma pa da ni sa). In Pada Varnas it is known as Mukthaayi Swaram-Sahithyam.
• Charanam: Sung with lyrics
• Chittai Swarams: Sung completely with syllables. In a Pada varnam, there are lyrics which correspond to the Charanam swaras. The swaras occur in several groups or stanzas.
Generally, a varnam is sung as follows:
• Muktayi Swaram
• Pallavi (in double speed)
Repeat, then Pallavi sung in triple speed, or in original speed.
• Charanam Swara Group 1
• Charanam Swara Group 2
• Charanam Swara Group 3
• Charanam Swara Group 4
There are generally 3–5 swara groups in every varnam. In a concert, the entire charanam section is sung at approximately 1.5 speed. Sometimes when repeating the Pallavi the Annupallavi and Muktayi Swarams are repeated in double or triple speed.
Varnams are generally sung in two varieties of talas, or metric systems, Adi Tala (8 beat cycle) and Ata Tala (14 beat cycle), where Ata Tala varnams are generally more complicated and advanced. In most of the Adi Tala Varnams the tala is placed in the 2 kalai version. So therefore each beat and finger count is placed twice.
Adi Tala Varnams include:
• "Sami Ninne" in Sree Ragam composed by Karur Devudu Iyer
• "Ninnukori" in Mohanam ragam by Poochi Srinivasa Iyengar
• "Evvari Bodhana" in Abhogi ragam by Patnam Subramania Iyer
• "Valachi Vacchi" in Navaragamalika (9 ragas, similar to Ragamalika which literally translates to a garland of ragams.
Ata Tala Varnams include:
• "Viriboni" in Bhairavi ragam by Pacchimiriam Adiyappa
• "Nera Nammithi" in Kaanada ragam by Poochi Srinivasa Iyengar
• "Chalamela" in Sankarabharanam by Swati Tirunal
The Arabhi ragam varnam is supposed to be the only longest piece with jantai and tattu prayogas.
The Valachi vachi varnam is a Navaragamalika composition.
The "MATHE MALAYADHWAJA" is a very Unique Varnam due to the fact that the Chitte swara has two different types of sahitya arranged according to it. The first is a rhythmic poetry describing the king and has the Raja mudra and the second is a liting Solkattu swara which is very nicely embodied in expression by Bharatanatyam dancers.