Harmony means playing several notes together to make “chords”. The word comes from the Greek harmonia meaning "to join things up" . A tune by itself can sound nice, but it can be “harmonized” by adding an accompaniment of chords. Studying how to do this is called harmonization. Music students learn which chords sound nice after one another. These are called “chord progressions”. Many music theorists have written books about harmony.
Music which is made of a tune with harmony underneath is called “homophonic”. In a way it is the opposite of polyphonic which means that each part (each voice) is a tune in itself. However, even polyphony needs to make pleasant harmony. Harmony as we know it in European music had become fully developed by the Baroque period (17th century).
One can play a chord with three notes using the 1st, 3rd and 5th notes of the scale of the whatever key the music is in. This gives a chord which sounds like the “home chord”. This means that at least three notes are needed for harmony. In most homophonic music there are four: for example a choir will normally divide into soprano, alto, tenor and bass, or a string quartet will divide into violin 1, violin 2, viola and cello.
Harmony which uses just the notes of the key (e.g. just using white notes for C major) is called “tonal harmony”.
Harmony which adds lots of extra sharps and flats is called “chromatic harmony”.
If music is not in any key at all, like in some music by Arnold Schoenberg, it is “atonal”. Harmony can be atonal.
Rameau's 'Traité de l'harmonie' (Treatise on Harmony) from 1722.
Harmony for Kids. Kiddle Encyclopedia.