There are a number of ways to think about harmonies. The most basic way is for them to essentially parallel the melody. That is, they have the same duration as the melody notes, and move in the same direction of the melody. The trick is to select notes that sound good with the melody.

A couple of rules of thumb to keep in mind:
  • Notes of the chord sound good.
  • Intervals of thirds and sixths sound good.
  • Intervals of fifths and octaves sound good, too. But thirds and sixths in a row sound better. (More on that later).
  • In two-part harmony, have both a "basic tone" (root or fifth of the chord) with a "guide tone" (third or seventh of the chord).
A "guide" tone differs from a "basic" tone in that it tells something about the quality of the chord (major/minor/dom7, etc.). Having both a "basic" and "guide" tone gives you a good harmony because even with only two notes, you still have the essential quality of the chordal harmony.

A simple way to fulfill all the above is to select a chordal tone that lies above or below the melody note, and place it below the melody note. For example, if the melody note was a "C" and the chord was Am7 (A-C-E-G), you could select an "A" (since it's the pitch below the C in the Am7 chord) or an E (the pitch above the C in the chord).

A caveat to the above rule: Don't use the 7th of a Maj7 chord, or you might end up with a minor second in the harmony, which is pretty dissonant. In general, you want to avoid intervals of a second in the harmony, except for brief passing dissonances.

Every melody note doesn't have to be harmonized with a chords. Notes of short duration (less than a quarter note) or that don't fall on a strong beat (beats 2 or 4 are "weak" beats) create passing dissonances which don't have to be harmonized. Harmonizing every note gives a very "fussy" arrangement that sounds more like a church hymn than a pop song.

Along those lines, try to keep a "common tone" when the harmony changes. For example, if you've got an "F" chord (F-A-C) and a Am7 chord (A-C-E-G), choosing the "A" or "C" note will work for both chords. So if you choose that as a harmony note, you can stay on it when the chord changes.

Another way to think about harmony is as a counter melody to the melody. Some general rules of thumb for counter-melodies:
  • When the melody is busy, the counter melody should be simple (longer note durations than the melody) - and the other way around, too. This keeps each part clear.
  • In general, the counter-melody should move in the opposite direction of the melody. This makes each part independent of the other. Obviously, they two will move in the same direction at times (usually in intervals of thirds and sixths). But the more they move in opposite motion, the more they will sound like counterpoint.
  • If you use an interval of a fifth or an octave, avoid following it with another interval of a fifth or octave. This is called "parallel fifths" or "parallel octaves". These are both strong harmonic intervals, and using two of these intervals creates such a strong harmony that it destroys the feeling of an independent line.
A counter-melody with no dissonance (all thirds and sixths) tends to be pretty bland, so the introduction of some dissonant interval that resolves adds a welcome relief. The most common way for this to happen in via a suspension. This happens when one of the two notes changes, creating a dissonant interval. The note that didn't change then moves stepwise (typically downward) to resolve the dissonance.

For example:

E-C "E" on top - consonant interval of a third
E-B "C" moves down to a "B", creating a dissonant interval of a fourth
D-B "E" moves down to a "D", creating a consonant interval of a third

It's called a "suspension" because the one voice changes, leaving the other "suspended" until is resolves downward. It's a very effective device, and you can have chains of suspensions. (But don't overdo it!) Like a lot of things in music, these are often best worked out backwards.

As with almost everything else in music, simple is usually better. It's more effective to have less harmony, and put in contrast with monophonic sections.

That's the theory, in a nutshell. Being able to sing it... that's a different matter.