How do sprites work?

I've seen sprites from old school games like Super Mario Brothers, and wondered how they're animated to make a game.

They're always presented as one big image map, so how are they used?

For Mario (as an example) are there precalculated image co-ordinates that outline mario, and are swapped between various mario sprites to produce animation?

Or are sprites pre "cut" during game initialization using precalculated images co-ordinates and stored in memory somewhere?

Obviously I know nothing about game development.