If you are unfamiliar with how musical intervals work, please go read How Music Works: Harmony
Chords are, quite arguably, the most important piece of modern music. Everything you hear is built around chords. But exactly the way they work may not be as clear.
In a major or minor chord, there will be three notes. Your root note, your third, and your fifth. Whether the third is a major or a minor third. For example. The image below on the left shows the notes played in a C major chord, and the image on the right shows the notes played in a C minor chord. Notice that the only change is in the third (middle) note.
Chords, however, are much more complex then that. For example, the addition of a Major 7th (11 notes above) to the C chord makes it a Cmaj7 chord (left), whereas the addition of a minor 7th would form a G7 chord (right).
Using the various other intervals you can create an almost infinite number of chords, although only some of them are recognized as actual “chords.”
Another thing to know. Just because you are playing a C chord, it does not mean that the bottom note of your chord has to be a C. As long as you are using those three notes (C, E, G,). Chords that do not have the root note as the lowest note are called inversions. In the illustrations below, both are C chords even though they seem different.
In other instruments such as guitar, the chord will still contain only those notes. Because it has six string though, a guitar will have multiple strings playing the same note (for the most part) in different octaves. This is one of the things that makes a strummed guitar chord sound richer then just three notes played on a piano.
All modern music is built around chords in some way or another. Without chords, we would be lost. We would be randomly playing notes. Chords give us a framework from which to build the rest of our music. Within that framework we are free to do whatever we want. Which is why two pieces in the same key using the same chords can sound so vastly different.