A big flat box covered in knobs, lights, buttons, sockets and sliders. Designed to accept m input signals, and manipulate and mix them together into n output signals, where m is usually greater than n. Also known as a mixer.
Sometimes they contain their own power amplification to directly drive loudspeakers, but usually they plug into external amps.
Looks much harder to use than it actually is. See my Guide to using a mixing desk if you need to know how to use one.