As the name implies, varNam means "description". In other words a varNam describes or brings out the svarUpa of the raga. It has the three parts of a Karnatak song viz. pallavi,anupallavi and caraNa. Generally the text is very short in nature and there is a main set of svaras known as the muktAyi svaram. After the caraNa there may be several groups of svarAs called ettugaDa svarAs. Normally the composition is sung with an AlApanA.
In a concert it is usually the opening piece and it is also supposed to be an exercise for the artist to have his/her vocal cords warmed up and prepare him/her for the rest of the concert.
Some varNams are long and elaborate. VarNams used for a dance also have sholkaTTu svarams.
Corrections welcome.