What exactly are chords, melody and harmony? How do you distinguish one from another? What are some examples of these being used in a DAW like Sonar?

Ignore me, proper answers appearing below..

Look for the documentary "How Music Works" and you'll get an excellent answer to all those questions. It's a BBC documentary which hasn't been released for sale or rent, so it's on YouTube apparently without issues from the BBC to remove it. There are separate sections on rhythm, melody, bass, and harmony. They are all quite good, entertaining and educational. The host, Howard Goodall, has a number of BBC documentaries on music that are quite good.

Harmony is essentially the chords ( a set of simultaneous notes) that support a melody which can be defined as the focal point of a piece of music.

Bibs

Technically speaking a chord is 3 or more notes at once. Two notes is described as an interval. Two notes alone don't constitiute a chord. Two notes (by themselves) don't have the ability to spell out a minor or major tonality.

The word harmony can have two meanings. One is to describe the chord progression. Harmony can also mean another melody that works in conjunction with the main melody. Harmony (in this sense) usually will have exactly the same rhythm as the melody and it is spaced an interval away from the melody. This interval has to be adjusted to fit the chord progression as the melody goes along.

Another secondary melody that works in conjunction with the main melody can be described as a counterpoint melody. But in this case the rhythm of the counterpoint is not necessarily following the exact rhythm of the melody.

I am not sure that I would agree that the harmony alone (chords) is the focal point of the music. (when both chords and melody are present) I think that the melody of the music could be described as the focal point of the music. It is really a combination of both chords and melody that gives us the strongest outcome. Remember a chord progression on its own in not a copyrightable thing, but the melody is. A melody on its own would stand up compared to a just a chord progression on its own. But the best and greatest and most emotional impact usually happens when both the melody and the chords are present.

Please also accept that I am not such a traditionalist that music has to have chords and melodies for it to exist and be good. Music can have any combiniation of these things and even just the percussive rhythm component alone as well. Music can be just a series of sounds if you want it to be. But good melody writing over well chosen chords is hard to beat in many ways and as I said before it is often the case when the music is emotionally effective. (Listen to Steely Dan for some serious melody over chord progressions.)

Sorry, I actuall meant that the melody is the focal point, not the harmony. If you imagine a picture of a bridge, the melody could be described as the horizontal part, whilst the harmony is the vertical structure that supports it. 
Obviously harmony, melody ,counterpoint, form or any other variations and terms can be discussed in much greater detail, but this depends also on the style/period of music to which the OP is referring.

I agree with Jeff that in modern music a harmony is usually meant as maybe an interval sung/played over or under the main melody with the same rhythm but I wouldn't describe two notes together as purely an interval as described as a chord it's known as a dyad, a three note chord a triad.
Cheers

Bibs

Rob brings up an interesting point about the dyad because it can imply or suggest  a chord type in its own way. eg the notes C and E may suggest or imply a C Major chord and the notes C and Eb may suggest a C Minor chord. It is not as strong as a three (or more) note chord but an interesting type of chord none the less. I am still correct in a way because two notes are a vague sort of chord but a strong interval.

What is also cool is the just two notes can suggest a C Major chord which is the I chord in the key of C, the IV chord in the key of G and the V chord in the key of F so with just two notes three keys can be implied as well.

It is also interesting how a single bass root note plus a two notes above that can spell out all 4 seventh chord types and a bass note plus three notes above can spell out many many chord types. It sounds good when you are economical or use a minimum number of notes to spell chords. Many musicians use far too many notes at the same time to spell out chords and the music is cluttered as a result. You don't really need a guitarist and a keyboard player to play the same chord at the same time, better they do it at different times but also even better when economical note choices (and different voicings) are used. Even better again when they are doing two different jobs or roles. (This is when the playing impacts on engineering. When too many people are playing too many notes in chords at the same time no matter of engineering will be able to separate them out as much as getting all those people doing different things then it is much easier to keep them separate in a mix. )

A great book for economical note choices for chords is 'Jazz Rock Voicings For the Contemporary Keyboard Player' by Dan Haerle. (for keyboard players at least)  What I found interesting while studying Jazz harmony is that they tell you all the notes that are piled up to spell out many chord types then they tell you to start leaving them all out and use the least number of notes to spell out chord types. It sounds way better and clearer. Also if you are using say only two (or three) notes to spell out chords (while the bass player plays the root notes) you can get through faster and harder chord changes because you only have to move two or three fingers around (and not by much either) to cover those changes easily.

Thanks Droddey here's a link I found: 
http://www.youtube.com/user/timegrinder

Jeff, excellent post. I remember the days of scoring 4 part harmony, no consecutive 5ths etc and the rigid rules composers had to follow. Fast forward to the pop era and everything changed. The skill of using less as more is even something I forget in the name of pop and this is an excellent reminder. 
Certainly, if the piece is in C major for instance, I don't need to play the 'E' or the 'G' if I want everyone to hear a Cmaj7. 
The best producers/composers  tell you want they want you to hear with the minimum fuss. Sure, double up for texture if needed, but otherwise keep the focal point the focal point and don't muddy it up with unnecessary notation. I know they say there are no rules in pop, but you have made an excellent point and one I will definitely try to adhere to

Cheers

Bibs

OK, I am starting to get a grasp of what these are thanks to some great feedback. Chords are clear to me now, just a combination of notes played together. So are chords those keys played with the left hand side of the piano or can chords also be those keys played with the right hand? What are some examples of harmony in music when heard? I'm still having a hard time distinguishing harmony from melody.