VarNams are sophisticated musical forms that include swaras and lyrics. For students, they are useful for learning the swaras of various raagas, singing in multiple speeds rapidly, as well as learning the appropriate gamakas. They introduce the proper combinations of swaras for each raaga and require great discipline for singing. The structure of the varnam includes pallavi, anupallavi, muktaayi swaram, caraNam and multiple ciTTa swarams that return to the caraNam (see the glossary). Varnams are usually the first pieces sung in concerts.
UPDATED! Here is an exhaustive (but not absolutely complete) list of varNams - NOW over 1400 of them! A few have lyrics and are linked. Full list provided by Lakshman Ragde.