Gyroscopes are like the less-flashy, workhorse cousin of accelerometers. They can be a bit tricky to define (check out a detailed explanation of the technology here), but for our purposes, let's simplify it by saying that a gyroscope is a wheel that rotates around an axis that has the ability to move. Because of Newton's first law of motion -- an object moving in a straight line will continue moving unless acted on by a force -- the axis will stay inert as the wheel's spinning rotations cancel out a force from one point or another. Thus, the wheel keeps spinning, and the axis keeps stabilizing.
While an accelerometer measures a change in velocity and can sense a change in orientation, a gyroscope is actually maintaining (or simply measuring) orientation. Because of their magical stabilizing abilities, gyroscopes are used in a number of devices that keep a large object steady (for instance, keeping a boat from rolling).
Ever wonder how an airplane's autopilot works? The gyroscope in the plane can keep a plane on a steady course by measuring the orientation of the plane relative to the scope. Add an accelerometer in -- which can tell us where the plane is going and its speed -- and you've got yourself an unmanned aircraft. (Gyroscopes are also used in drones and the like for the same purpose.) And don't forget your wireless mouse, which steadies your jerky hand motions by using a stabilizing gyroscope.