I have read that Burnside's theorem implies that a group with order $p^aq^b$ cannot be simple. So I looked up Burnside's theorem and saw that it doesn't mention "simple" explicitly, rather it says that such a group is solvable. Firstly, I am not sure I fully appreciate/understand the definition of "solvability". What is the motivation behind defining such a property and how does it imply non-simplicity in our $p^aq^b$ group?
Added: Could anyone please explain why "only finite solvable simple groups are the cyclic groups of prime order" (as suggested by Qiaochu) without quoting any particular theorem? This is probably easy to do, but I don't quite see it. Thanks.