I'm going to start with the very basics. Things can, of course, get more advanced than this, but its ultimately all based on what follows:
Every note in the scale has a corresponding chord that can be built on it. The starting note is called the "root" of the chord.
A chord is then built simply by "stacking" more notes on top of the root. Specifically, these notes should be (1) in the scale and (2) separated by an interval of a third from the previous note. A basic triad (three-note chord) is built by stacking two thirds on top of the root, a "seventh" chord adds an additional third on top of that.
For example, lets say that you're playing in the key of E major (a common guitar key, which includes the notes F#, G#, and C#). You can construct a chord on E by using the notes E then G# then B. You would construct a chord on A as A, C#, E.
One last thing to note: the order of notes in a chord, the number of times each one appears, and octave they appear in, is largely unimportant (this property is called the "voicing" of a chord). So any combinations of A's, C#'s, and E's would make a A chord. Most often, though, the lowest note will be the root.
As rlo notes, if you're playing in a major key, the chords that are formed on the 1st, 4th, and 5th notes of the scale will be "major" while the chords that are formed on the 2nd, 3rd, and 6th degrees of the scale will be "minor". This term refers to the distance (in half-steps) from the root to the first 3rd that you stack -- in minor chords, that distance is 3 half steps (e.g. A to C), while in major chords, that distance is 4 half-steps (e.g. C to E).