So you've written a memorable hook, or a nice little melody. You put together some chords to go with your creation and that's it. What a lot of musicians don't seem to realize is that with any given set of notes, there are a ton of different ways to harmonize it. If you've ever heard some of the mash-ups of well known songs put on top of other tracks, you can see how there are many things you can do with the background.
The Basics
Whenever you've written a melody, there are a set of chords or harmonies that we may 'automatically' hear. For example if it's a simple melody that doesn't move around much or have any weird leaps and accidentals, we will probably hear a basic
I IV V I. Since we hear these progressions so often, we may automatically hear them in our head. That doesn't mean that we have to use them, or even that they're 'right'. For example if you have a simple melody you may want to use other chords because you want a different 'feel' for the song. Or, you want to invoke some surprise, or you want to change it into a different genre, or you just may want to make the song 'more interesting'.