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OK. Evolution (in biology) means that a population changes over time. This change happens during the reproduction process of the individuals this population consists of. So you get offspring that has a mix of its parents genetic material plus some mutations which always occur to some extent. Some of these new configurations can produce individuals that are better suited to their environment than their parents or their siblings, so they have higher chances to procreate and on average they also have more offspring.
Over time the genetic makeup of this population can change considerably. So if you had a specimen from the current population and one from the ancestor population and compared them, it can make sense to assign them to two different species.
Of course a population can also split up, and the two (or more) groups get reproductively isolated. This means that changes that arise in one group are no longer shared with the other group(s). Usually these subpopulations are exposed to different environmental conditions which means that natural selection affects them in different ways.
If they stay isolated long enough, it can happen that they changed so much that they don't recognize each other as potential mates. This can be nicely observed in ring species where the two populations at the end of the 'ring' don't recognize each other (here is a nice illustration of a warbler ring species).

First, I thank you for posting a definition of what I am supposed to not understand. What you posted is exactly what I understood evolution to be postulating. This version varies of course slightly with other versions of how evolution allegedly occurred, but I think this is a pretty good summation. Thank you again.

OK. Evolution (in biology) means that a population changes over time. This change happens during the reproduction process of the individuals this population consists of. So you get offspring that has a mix of its parents genetic material plus some mutations which always occur to some extent. Some of these new configurations can produce individuals that are better suited to their environment than their parents or their siblings, so they have higher chances to procreate and on average they also have more offspring.
Up to hear I agree.

Over time the genetic makeup of this population can change considerably.
I fall off track here. Significant mutations of the kind postulated by evolutionists are not seen. The genetic makeup of an ape does not eventually over any number of years make a man. The specific "this is a man" element is missing from his make up.

So if you had a specimen from the current population and one from the ancestor population and compared them, it can make sense to assign them to two different species.
Species, being a man-made classification.

Of course a population can also split up, and the two (or more) groups get reproductively isolated.
No matter what variation in human population you see, a pigmy is still as much of a man as a North American scientist. You are not going to get the kind of change you are reaching for.

This means that changes that arise in one group are no longer shared with the other group(s).
According to the hypothesis which said that they shared reproductive capability before.

Usually these subpopulations are exposed to different environmental conditions which means that natural selection affects them in different ways.

If they stay isolated long enough,
How long is 'long enough'?

it can happen that they changed so much that they don't recognize each other as potential mates. This can be nicely observed in ring species where the two populations at the end of the 'ring' don't recognize each other (here is a nice illustration of a warbler ring species).
Assuming they were genetically fit to reproduce with the comparison species to begin with.