Where would you use a gyroscope?

Gyroscopes are used when you want to know how fast or how much something is turning.There are many ways to determine rotational rate (optically, magnetically, etc), but a gyroscope is unique in requiring nothing external to make this measurement. Here are a few examples:

In automobile electronic stability control systems the rotation rate of the car is measured by the gyro and compared to what is expected from the wheel speed and steering wheel sensors. If there is a discrepancy between them, differential braking is applied to bring the car back into control.

Optical image stabilization in digital cameras is accomplished by having a gyro measure the unintentional rotational movement of the camera. The image is then steered by moving lens elements such that it remains relatively stationary compared to the image sensor. In this way images remain blur-free even though the camera moved during exposure.

In navigation systems, a gyro's angular rate output is integrated to determine angular heading which in turn is used to determine if you have been turning and by how much. When combined with displacement information you can figure out your position.