What makes simple groups so special? The classification of finite simple groups was one of the most important problem in group theory. But what makes simple groups so interesting and special?
 A: Simple groups are like prime numbers........
One must not let the account of the matter end with a full stop after the last word above.  If one could say that all finite groups are products of finite simple groups, then the analogy would be simpler.  But one can say that simple groups are as far as you can take the process of taking quotient groups without going to the very smallest quotient group: the group with only one element.
The complication is that just taking Cartesian products of simple groups doesn't give all finite groups, and in fact, I think the understanding the ways in which other finite groups are built out of finite simple groups is quite a substantial problem in itself.
However.  Say you have a composite number like $299.$  You can form quotients:
$$
299/13 = 23.
$$
$$
299/23 = 13.
$$
But from the prime number $23,$ you can't form any quotients except $23$ and $1.$
Similarly there are no quotient groups of a simple group except itself and the group with only one element.
