In characteristic zero, the group algebra is semisimple, so there are finitely many simple representations.  These representations correspond to the blocks in the decomposition as a product of matrix algebras.  As for characteristic p, this is modular representation theory,  The number of irreducibles is the number of $p$-regular conjugacy classes (where $p$-regular means the period is prime to $p$).  See, e.g., Serre's _Linear Representations of FInite Groups._