Despite the name, simple groups are far from "simple". The finite simple groups are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way prime numbers are the basic building blocks of the integers. This is expressed by the Jordan-Hölder theorem. In a huge collaborative effort, the classification of finite simple groups was accomplished in 1982.
The only abelian simple groups are the cyclic groups of prime order. For these groups the term "simple" is appropriate. The smallest nonabelian simple group is the alternating group A5 of order 60, and every simple group of order 60 is isomorphic to A5. The second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and it's possible to prove that every simple group of order 168 is isomorphic to PSL(2,7).