There are 2 major classifications of Carnatic Ragas. They are..
1. Melakartha ragas : Arohanam : S R G M P D N S' & Avarohanam : S' N D P M G R S
These are 'kartha ragas' or mother ragas.
Eg., Shankarabharanam : Aro : S R2 G3 M1 P D2 N3 S' & Ava : S' N3 D2 P M1 G3 R2 S
Chakravagam : Aro : S R1 G3 M1 P D2 N2 S' & Ava : S' N2 D2 P M1 G3 R1 S
The total number of melakartha ragas = 72 (suddha madhyamam 36 + prathi madhyamam 36)
Details of Melakartha ragas can be viewed here : Melakartha Ragas
2. Janya ragas : These are ragas derived from melakartha ragas, by removing / jumbling some swaras either in the arohanam or avarohanam or both.
Eg : Mohanam : S R2 G3 P D2 S' - S' D2 P G3 R2 S
Hindolam : S G2 M1 D1 N2 S' - S' N2 D1 M1 G2 S
Among the Janya ragas, there are some classification, based on the number of swaras in the arohanam & avarohanam. While counting the number of swaras, 'S' should be counted only once..
7 swaras - sampoorna ragam
6 swaras - shadava ragam
5 swaras - audava ragam
4 swaras - svarantara ragam
In case there are 7 notes in the arohanam & 6 notes in the avarohanam, they are called sampoorna-shadava ragas. If there are 5 notes in the arohanam & 7 notes in the avarohanam, they are called audava-sampoorna ragas.
The total number of janya ragas = INFINITE
For more details on Janya ragas, please view : Wikipedia on Janya Ragas