I posted the following answer yesterday after only a quick skim of the question. When I read it with more care, it seemed to me to be the answer to a different question entirely. After having looked at some of the other answers, I find myself less sure of whether my answer is relevant or not! Anyway, here it is: make of it what you will.
I believe that a complete account of the Albert-Shimura classification of endomorphism algebras of complex abelian varieties may be found in Chapter 9 of Birkenhake and Lange (Complex Abelian Varieties).
As someone who has worked in the area of abelian varieties with endomorphism structure, I have always found the complete classification to be rather daunting. One particularly subtle point is that after you set up a moduli space of abelian varieties having endomorphisms at least by a certain order in a simple algebra, you are still left with the task of showing that a sufficiently general element of the family has precisely that order as the ring of endomorphisms and not something larger. But the discussion in Section 9.9 (I am looking at the second edition) successfully negotiates this subtlety and does, I think, in fact give a complete classification.
Good luck!
Added: for the superusers: the above is correct but is not an answer to this question. I didn't read it carefully enough until after posting the answer.