In western music, we have 12 tones.

A scale is just a choosen part of those tones. You can have 5 notes scales (pentatonics) up to 12 tones scales (chromatics) 4 and 3 notes are usually arpeggios, or chords. And under that, there's not much, so it's intervals.

Saying you play a scale over a chord is saying 'which set of notes do you use when playing over that set of 3/4 notes'.
Given the fact that music isn't exact, in the same situation, one will choose a set, while others will choose another one, while others don't have a clue because they don't know the corresponding set (or scale).

That's where mode theory start. Since a chord only has 3/4 notes, the scale you choose can be made with those 3/4 notes, but then, you can fill in the blanks. Depending on which notes you fill it in, you will play in a certain mode or another.