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.
The only simple groups which are abelian are the cyclic groups whose order is a prime number. In a huge collaborative effort, the classification of finite simple groups was accomplished in 1982.