A practical overview
Intervals is about the distance between two notes or keys. Keys are about what we see, also about sound. Notes can be what we play, but also what we see in notation.
Our seven note notation system uses seven letters, and every problem we have about how to write notes has to do with that. F G# and F Ab are the same thing on the piano. They both skip two keys, one white and one black. But because traditional interval names are based on letters, one is a 2nd, and the other is a 3rd. So now we have two problems:
First, are both correct? (They are.)
Second, how do we know which to use?
Actually, we have a third problem. If someone uses the wrong choice in music, how do we know? Or if we use the wrong choice, and something looks weird, how do we fix it? What rules do we use for guidelines? I hope the pages explaining each interval will clear this up.