How does natural selection relate to evolution? Is it something that is completely different than evolution? Can natural selection happen without evolution?
Natural selection is one mechanism of evolution. Suppose you have cows which due to genetic variation have different amounts of white and black spots. If you as a farmer were to go out and select only a few of the blackest cows to breed, then over time we would expect your herd of cows to tend toward being darker overall.
Since you as a farmer are doing the selecting that is "artificial" selection. "Natural" selection would be something like cow-eating wolves being able to eat the lighter cows more easily, which meant darker cows were more often able to survive to breed. Or maybe the lighter cows tended to get skin cancer more often and thus were not as prolific.
This sort of natural selection can happen without evolution occurring for a variety of reasons. For example if there wasn't any genetic variation in the population; if all the cows were genetically identical and never experienced mutations, genetic drift, transcription errors, etc... then the selection wouldn't effect anything.