The quick explanation is that a channel refers to a single path through which an audio signal can pass.
On a preamp, X channels means you can run X microphones (or other things like instruments, depending on its design) through it separately at the same time, and get as many separate signals back out.
On a mixer, the number of input channels refers to the same thing - number of separate signal paths that can run side by side. In this example, however, the output of the mixer may not (and very well does not) correspond to the number of inputs - most often you see mixers that have a lot of input channels (12, 16, 24, 32+ are just a few), that only output a stereo set of channels, left and right.
You mentioned studio monitors, but those are traditionally hooked up in a studio such that all the channels of audio you are utilizing are summed up somewhere, and then sent as a composed setof Left and Right channels (usually - sometimes you might be listening to something in mono, or in surround).
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