Well, you need to know all the intervals and chords that they create. At the very least, you should know how to make major and minor triads derived from the scale the melody is composed with.
You should also understand the difference between consonance and dissonance.
There is an endless number of ways to harmonize your melody. You could make each note a new chord if you wanted too. As Magellanic said, you should look for "chord tones" in the melody. For example, if the first bar of the melody primarily uses the notes A, C, and E, you could play an A minor chord beneath that.