Now, you've just written your next million seller, but something's missing: NO CHORDS! (Unless you have very good music software.) Harmony is perhaps the most complex subject in all of music, there are literally tens of thousands of different chords that you can use. I can't even begin that now. What I will do is...show you the basics!
The very first thing you have to do is find the "key" of your song. That is to say, the scale that it is based on (unless you've got that very good software, and you already know). If you have a keyboard, or some other instrument, get a book of scales and try playing a few of the major scales, unless you can tell from the sound, that your melody is definitely minor. If you can hear most of the notes of your melody in any one scale, this is probably the key. Alternately, you can have a musician friend do this for you. If your melody is in a mode or uses a jazz scale (i.e. diminished, altered etc.), you don't need to be reading this! Now, for the last step.
You're almost done now, simply take chords made from the first, fourth and fifth notes of the scale related to the key your song is in, and try them in different combinations. That is to say, if your song is in the key of "C major" your chords are C, F, and G. This is very basic and will probably give even someone who is not a musician only limited satisfaction, but this is enough to give you a start, and this is all that I will touch upon with regards to harmony, since some of the biggest songwriters of all time were not musicians, and knew no theory at all.