Chords are produced when you play three or more notes simultaneously, or one after the other in quick succession ("arpeggios"), forming interesting interval relationships.
The purpose of playing chords is to produce harmony. Some chords produce a pleasant "harmonious" sound, others produce tension and "dissonance", asking to be resolved later.
Some important chords, called "triads", are produced by playing only three simultaneous notes.
The main triads are built by playing a root note, plus a third, and, a fifth interval notes. This combination produces a "consonant", or "pleasant" chord.
Some examples of chord triads, using C as a root reference (may be transposed to other root notes, or inverted):
[For the explanations, we use here the piano, because it is the easiest way to see all the notes in a linear position]
"Quads", are chords produced by playing four simultaneous notes.
A Quad plays the three notes in a triad, and adds a fourth complementary note.
Quads produce a "dissonant" sound, usually used as transitions between consonant chords, producing alternations of tensions and resolutions.
Some examples of chord quads, using C as a root reference (may be transposed to other root notes, or inverted):
It is also possible to produce more complex chords called "extended chords", which contain 5, 6 and 7 different notes.
These are the ones usually marked as "9th", "11th", "13th", etc. They are built at 9,11,13 intervals (respectively) from the root.