Multi-channel amplifiers are power amps that can drive a number of loudspeaker channels. Unlike stereo amplifiers which have 2 channels (left and right), multi-channel amplifiers can drive 4, 6, 12, or even 16 separate speakers. This could be different zones in a multi-room audio system, or perhaps a multi-channel surround sound setup.
Multi-channel amplifiers should not be confused with multi-room amplifiers. The latter is an amplifier with a built-in control system and multi-room audio capabilities. Whereas a multi-channel amplifier has no control system and is purely a power amp for a wider system.