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Picture a spinning gyroscope that has its axis of rotation tilted with respect to gravity. As the gyroscope spins, the direction its axis is pointing is moving. It is describing a circle. That's called precession. Gravity is pulling down (applying a torque) on the gyroscope and trying to pull the top over. Gravity is trying to change the alignment of the axis of rotation. But because the gyro is spinning, because it has what is called angular momentum, gravity can't pull the top straight down. Instead, the force acts 90 degrees later in the direction of rotation. The "down" force is combined with the "spinning" force to create a "sideways" force. The result is that the axis of rotation moves a little bit to the side. In the next instant, gravity, which is still pulling, tries to bring it down, but the spin, that angular momentum, again combines with the gravitational force and another bit of "sideways" force shifts the axis again. In each instant of time, the force of gravity and the gyroscope's spin result in a dynamic that constantly shifts the axis or rotation a bit more to the side. Momentum is conserved in this phenomenon. In an experiment, a top spinning in one direction and tilted as it spins will exhibit precession in a given direction. If the top is spinning in the other direction, precession will be the opposite that it was with the first top. The Wikipedia article on precession has a picture of a gyroscope that has a bit of animation to (hopefully) make it clearer.

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Q: Why does a tilted gyroscope not fall?
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