This term is often used as a synonym to distortion.
Overdrive occurs when input gain exceeds the capacity of a device to handle the amount of gain thrown at it, for instance a tube inside an amp. The signal then gets clipped and that creates a distortion effect.
This could also be explained as distortion being the result of overdrive or over-powering an amplification device. Thus we could also look at the matter in terms of "before" and "after", being the overdrive what we put in, and distortion what we hear.
But there are also many stomp boxes labeled as overdrive and distortion, and this certainly creates some confusion, as both of them are placed before the amp. Regarding stomp boxes we could say that the main difference between an overdrive and a distortion is that overdrive stomps are more subtle, and use soft-clipping circuits, while distortion stomps use hard-clipping circuits which affect the signal greatly, with great clipping.