Play three or more notes at once and you have a chord. Western harmony builds them by a simple recipe: take a note from the scale, skip one, take the next, skip one, take the next.
Triads
Stacking two thirds gives a three-note chord, a triad, and it is the backbone of almost all popular and classical harmony. Built on C from the major scale, you get C, E, G.
The quality of the triad depends on the sizes of the two thirds involved.
A major triad has a major third on the bottom and a minor third above. It sounds bright and settled. A minor triad reverses them, and sounds darker. A diminished triad stacks two minor thirds and sounds unstable and anxious. An augmented triad stacks two major thirds and sounds unresolved and dreamlike.
Two of these four account for most of the music you have heard.
The chords of a key
Build a triad on each degree of the major scale using only notes from that scale, and the qualities fall out automatically in a fixed order: major, minor, minor, major, major, minor, diminished.
They are numbered with Roman numerals, capitals for major and lower case for minor, so a key gives you I, ii, iii, IV, V, vi, and a diminished vii. The three majors, I, IV, and V, are the most stable and appear in an enormous share of folk and popular songs. Add vi and you have the four chords behind a genuinely large fraction of the last seventy years of pop.
Because the numbering is relative to the tonic, a progression written as I, V, vi, IV works in any key. That is why a musician can call out "one five six four" and a band that has never rehearsed together can play it.
Sevenths and colour
Add one more third on top and you get a seventh chord, with four notes instead of three. The added note introduces a mild dissonance that gives the chord somewhere to go.
The dominant seventh, built on the fifth degree, is the most important of these. It contains a tritone between two of its notes, and that restless interval strains toward resolution onto the tonic chord. This is the strongest cadence in Western music, and it is why the V to I move sounds so much like an ending.
Jazz keeps stacking, adding ninths, elevenths, and thirteenths for chords rich enough to carry a whole texture. The principle never changes: thirds piled up, with each addition trading stability for colour. What ties a series of chords together is the key.