How to tell if a proof relativizes?
A proof relativizes if every step in the proof relativizes. You should go over your proof, and check whether each step remains valid in the presence of an oracle.

In contrast, a result relativizes by definition if it still holds relative to an arbitrary oracle. Hence, to show that a result doesn't relativize, all you have to do is give an oracle with respect to which it fails.

If we're interested in some specific statement, whose truth value we are not sure of, and there are both an oracle relative to which the statement holds and an oracle relative to which the statement doesn't hold, then we can conclude that no relativizing proof can either prove or refute the statement.