What are the differences between a canyon and a gorge?

A canyon is a deep valley with steep sides, think of the Grand Canyon most famously to visualise what this is.

A gorge is a deep ravine, which usually has a river running through it - though this doesn't have to be the case.

Finally a valley is any depression, usually of a certain length, in the surface of the land and often contains a river.

So, a canyon is a specific type of valley, with particularly steep sides. A gorge is a particularly deep depression, and will usually contain a river.