Varnams
The student of Carnatic Music starts with Sarali Varisai (solfa in different tempos) Janta varisai (solfa of paired notes), Dhaattu varisai (note practice with jumps), Alankaaram (solfa in different thaalams), Geetham (simple songs without sangathis or difficult phrases) and then goes on to Varnams. Varnams are said to epitomize the grammar of the raagam (raaga lakshanam). Varnams are of 2 types, the thaana varnam and the padha varnam. The thaana varnam consists of a pallavi and anupallavi followed by "mukthaayi swaram" and then by a single aavartham of charanam again followed by 3 to 6 chitta swaram groups, with increasing number of aavarthams. The lyric of varnam is sparse in words and gives training to the voice in "akaara" or melody without consonants. Concerts are often started with a varnam. Little is known of the origin of the word 'thaana varnam' but one school has it that the note groups (usually threes, twos and fives) are like thaanam singing and this is stressed when varnam is played on the Veena.
The padha varnam is meant for dance but is also sung in concerts (ex.'Nee indha maayam' in Dhanyaasi). It has a structure similar to the thaana varnam but words of lyric are also available for the mukthaayi and chitta swarams. The single line of the charanam often gives considerable scope to the Bharatha- naatyam dancer to portray a number of legends associated with the main deity on whom the varnam is composed.
Swara Jathi
Though the term "jathi" would imply the use of rhythmic mnemonics used in dance like 'tha thi thai thom' etc. Swarajathi as sung now a days do not use these jathis but consist of a pallavi, anupallavi and a number of charanams in which the the swarams (solfa) are first sung followed by the lyric. Swara- jathis of Syaama Saasthri in Bhairavi and Thodi are well known. Simpler swarajathis in Bilahari and Aanandhabhairavi are usually taught to students before teaching varnams.
Thillaana
Thillaana is a lively form of composition making extensive use of the dance mnemonics. The compositions are a mixture of these jathis, lyrics and swarams. Essentially composed for use in Bharatha natyam, thillaanas are also very popular in musical concerts, the jathis providing scope for complex rhythmic patterns.
Padham and Jaavali
Padhams were originally simple devotional songs, but with the adoption of padhams extensively for dance later compositions included the 'Naayaki - Naayaka' concept of devotion with romantic passages. Jaavalis are explicitly romantic and also used for dance. Padhams (especially those of Khsethrayya) are sung both in ghana raagams like Thodi and rakthi raagams like Sahaana. Jaavalis are generally set to rakthi raagams.