The maximal subgroups of $A_n$ are given by the O'Nan-Scott Theorem. They lie in one of the following classes:
1) $A_n \cap (S_{n-k} \times S_k)$, that is the stabiliser of a $k$-set.
2) $A_n \cap (S_a wr S_b)$ where $n=ab$, that is the stabiliser of a partition.
3) $A_n\cap AGL(d,p)$ where $n=p^d$ for some prime $p$.
4) $A_n \cap (S_m wr S_k)$ where $n=m^k$, that is the stabiliser of a cartesian power.
5) $A_n \cap (T^{k+1}.(Out(T) \times S_{k+1}))$ where $n=|T|^k$ and $T$ is a finite nonabelian simple group
6) an almost simple group acting primitively on $n$ points.
For classical groups the main result is Aschbacher's Theorem in the paper pointed out by Rivin. More details of the structure of the subgroups is given in the book by Kleidman and Liebeck.
For exceptional groups of Lie type there are papers by Liebeck and Seitz as noted by Barnea.
Both these results and Aschbacher's Theorem have the same philosophy as the O'Nan-Scott Theorem, namely that a maximal subgroup is either one of a small number of natural families that are usually stabilisers of some geometric structure, or is almost simple.
For sporadic simple groups, all the information in in the online Atlas. They are all known except for the monster. In this case there are a couple of possibilities for maximal subgroups but where it is not known if they are actually subgroups.
