Here is pretty much a simplistic view of how evolution really works.
I may be a little off in the examples but this is a general idea.

It can come about in a few different ways "evolution" can happen.

So take the Galápagos Finches they are all pretty similar with the only difference being beak sizes. So there are a buncha finches all with the same size beak and life is rough for the finches, they all eat the same size nuts so finding food is kinda hard. Well one day through a mutation a finch is born with a bigger beak than normal, he can eat larger nuts the other finches cant eat. Because of this he is able to get strong and knock up a buncha finches and have a buncha babies who some will inherit his mutation of a larger beak. And after time two types of finch emerge ones with "normal" size beaks and ones with larger beaks.

Another type of evolution is were one species decides to break into different groups. So say there is a group of fish of one species. The group gets pretty big and half the group leaves to try to find better food. Well over time the two different species will have different genetic mutations and these genetic mutations could be different between the two groups and eventually after a long while there are so many differences that they become two different species.

there are a lotta different circumstances that lead to evolutions but the basis is just mutation occur on the genetic level and get passed along from generation to generation.

/ramble