A circuit is a path that electrons take to do some kind of work with electricity. Circuits have a power source, like a battery, and must form an unbroken path back to "ground," which can be the other terminal of the battery or even a metal rod stuck into the ground. A circuit serves some purpose, like taking an input (such as the flip of a switch) and converting it to an output (like lighting a light bulb).
For a circuit to do work, it must be an unbroken pathway from power source to ground. Circuits that take power and feed it back into the power source without doing any work are called a "short circuit"; circuits that have a break and thus won't send electrons all the way through are called an open circuit.