I've never worked with a VCS, so please correct me if I make any fundamental mistakes.
For a project of mine, I've chosen to use Subversion, and I've been reading the documentation. If I understand correctly, the revision number is incremented for every check-in. However, this brings up a question. Is it possible to checkin more than one file at a time (so the increment is only one)? Also, what if, say, a checkin doesn't compile... will the revision number be incremented?
Is it possible to checkin more than one file at a time (so the increment is only one)?
Yes, this is possible. See Does Subversion have Changesets? from the official FAQ.
Also, what if, say, a checkin doesn't compile... will the revision number be incremented?
You would usually test first whether the current state of things compiles in your testing environment, and check in if it does.
You could automate this using a pre-commit hook that runs your tests, and on that condition continues with the check-in.