Locality is the idea in physics that things have to directly interact with other things in order to change them, e.g. a billiard ball striking another billiard ball. Until the 1900's, locality was the only line of thought. Even the "forces at a distance" like gravity adhere to locality because a gravitron, or gravity wave, or however you'd like to think of it, leaves mass A (a local interaction) and hits mass B (a local interaction) causing it to accelerate.
Non-locality is the idea that things don't have to directly interact in order to affect each other. For example, if a photon becomes "entangled" with another, interacting with that photon will cause a change in the other, instantly, no matter where in the universe it is. For more information, google "quantum entanglement".