So what is a gyroscope?(The following definition of a gyroscope is simplified)
Usually gyroscopes take the form of a disc shaped object but can be any object that produces an effective gyroscopic behavior. Most of the gyroscopes mass should be as far away from the center as possible. This often results in a disc with a large heavy rim. When the gyroscope isn't rotating it behaves like any other object, however when the gyroscope is spun on its axis at high speed it resists movements in certain directions.
When the gyroscope is spinning it can contain large amounts of stored energy. Newton's first law of motion states that any body will continue in its state of motion (still or travelling) until outside forces change it. Because of the forces when the gyroscope is spinning, if it is moved the gyroscope will try to 'compensate' for this movement. If we take the example of a gyroscope's axle being clamped to the structure of the car (often the cars engine behaves as a gyroscope because of its shape, mass and rotations).
If the gyroscope/engine is spinning at a fairly high speed in a clockwise direction, as seen from the back of the car and as shown in the diagram above. When the car is turned to the right (clockwise from above) forces A and B are applied to the structure of the car forcing the front end of the car down and the back end up. If the car is turned to the left then the front end of the car is forced up and the back end forced down. If the gyroscope is spinning in the opposite direction then the reverse will happen.
However if the car was moved directly upwards, downwards, forwards, back-wards or side-to-side the gyroscope would not apply any extra forces. The gyroscope only applies extra forces when the car is moved at an angle (turning left/right or when the front/back of the car is moved up/down). This effect is known as gyroscopic precession.
More about precessionFigure 1 Figure 2 Figure 3
The Figure 1 shows a bicycle wheel acting like a gyroscope being supported at both ends of the axle, if left the gyroscope would just slowly stop spinning. However if the right support is removed then gravity exerts a force pushing the right hand side of axle down. As described in earlier, gyroscopic precession will force the wheel to precess around its axis, as shown in figure 2 and 3 by the red arrows. The direction the precession takes depends on the rotational direction of the gyroscopes, shown in the diagram as green arrows. If the wheel is continuously unsupported then the wheel will continue to rotate around its axis (the pole).
The animation below shows a gyroscope precessing round a tower (shown in red). The gyroscope is shown in brown with the frame/cage/bearings in yellow.
Having fun with toy gyroscopes/spinning tops
Note: A spinning top is just a gyroscope.
The first picture shows a toy gyroscope being held by a piece of string looped round one end of the gyroscope. You should just be able to make out the string and my fingertip. As you can see the gyroscope stays level, almost in the exact position that it was started from and not dangling or falling off the string loop as you may expect. What you can't see is the fact that the whole gyroscope is rotating around the string due to precession. As the gyroscope slows down it starts to lean over and eventually falls of.
The next set of five pictures (Top left is the first, bottom right the last) shows the effects of the gyroscope rotational speed slowing over a period of time. As it does so the gyroscope slowly changes its angle to the floor (Leaning over). Note it is also precessing around its axis (Looking from above, going round the base clockwise).
For a more detail explication on how a gyroscope works click here.