Two squirrels fall out of a tree. One survives because it had a mutation for a very small flap of skin on its side that slowed its fall. The other...

This squirrel with the mutation for the small flap of skin will now have increased biological fitness. Fitness in this sense does not mean how strong or fast it is, but it refers to being able to survive and produce more offspring. The squirrel will pass on the mutation for the small flap of skin to its offspring, and this trait will become more common in the population by the process of natural selection. Over time, the squirrels with slightly larger flaps of skin will likely survive even better than those with smaller flaps of skin, and so the genes for larger and larger flaps of skin will get passed on more frequently. This is a very gradual process happening over many generations. Throughout the process, the squirrels that live during the in-between time as the two groups (with and without flaps) are in the process of separating into two species will be referred to as transitional forms. 

To become two separate species, the two groups of squirrels must somehow begin to mate with only each other. Perhaps the group of squirrels with the large flaps of skin begin to live higher up in the trees because they can now survive the fall and become separated from the group of squirrels without the flaps of skin who continue to live closer to the ground. However it happens, in order for them to become two different species, the groups must become reproductively isolated, that is, only mate with squirrels in the same group. When this happens, the gene pools of the two separate groups will evolve on their own without mixing and two separate species will result.