A group algebra over the complex numbers, like any semisimple algebra, is isomorphic to a product of matrix rings,
$$R = M_{d_1 \times d_1} \times M_{d_2 \times d_2} \times \cdots \times M_{d_n \times d_n}$$
The d_i that appear are exactly the dimensions of the irreducible representations of G. I don't know how to classify all sets of numbers that appear in this way (but the answer is not "all of them").

Over nonalgebraically closed fields of characteristic zero, nontrivial divison algebras can appear in the group algebra. I don't know if there's a restriction on the division algebras that can appear.

The situation in characteristic p is more complicated, but we can say something. A finite-dimensional algebra A has a square matrix of numerical invariants called the Cartan matrix. (I am not talking about the Lie-theoretic Cartan matrix, but I would be interested to know if they are named the same for a reason.) There is a one-to-one correspondence between simple A-modules and indecomposable projective A-modules, and the ij entry in the Cartan matrix is the Jordan-Holder index of the ith simple module in the jth projective module.

The theory of Brauer (and the subject of "part 3" of Serre's famous book on representation theory of finite groups) imposes strong conditions on the Cartan matrix when A is the group algebra of a finite group (over a large finite field). It must admit a factorization as D.D^t, where D is another matrix with nonnegative integer entries. (D is the "decomposition matrix" which describes what happens to simple modules in characteristic zero when reduced mod p.) For instance the Cartan matrix must be symmetric.

What if we work over a ring that is not a field, e.g. the integers? Here's a comment on Yemon's point that there are many pairs of groups G and H for which the group rings C[G] and C[H] are isomorphic. It is more difficult to construct such isomorphisms over smaller rings, and whether or not an isomorphism of the form Z[G] = Z[H] implied that G = H was an open problem for a long time (the "isomorphism problem for integral group rings," posed by Brauer in the 60s) A counterexample was found by Hertweck 10 years ago:

http://www.jstor.org/pss/3062112

(Pete points out above that I am assuming G is finite. When G is infinite C[G] cannot be analyzed by Wedderburn's theorem, there's no such thing as a Cartan matrix, everything breaks down. Is there a counterexample to the isomorphism problem simpler than Hertweck's if we do not require G and H to be finite?)

essential image. In the same vein, a functor F: A → B such that every y ∈ B is isomorphic to F(x) for some x ∈ A is calledessentially surjective. – Alberto García-Raboso Jan 31 '10 at 18:31