So, gyroscopes seem really complicated, but the principles that make them function are really quite simple. The short answer is: momentum will be conserved.
So, let's take the gyroscope, not spinning, and place it on the end of the stick, sideways. What happens?
It falls over, of course. But what else happens? Well, the top of the wheel gains momentum moving outward, and the bottom of the wheel gains momentum moving inwards. This is purely due to the acceleration of gravity on the gyroscope.
Now, we give it a good spin and do the same thing.
The same forces are acting on the gyroscope. The same acceleration of the top and bottom of the wheel occurs. But now some other action is in play. The part of the wheel that was moving inward toward the stick is now on the side of the wheel, not the bottom. Also, the part of the wheel that was moving outward is on the other side of the wheel, not the top.
The momentum obtained by the acceleration of gravity must be conserved, so instead of accelerating the gyroscope downwards, it accelerates it at a 90 degree angle. That's why the gyroscope spins about the stick you place it on. It can't just sit still, because that momentum has to go somewhere. This is called gyroscopic precession.
However, let's say you had a small motor with some nifty bearing conductors on the stick that could keep the gyroscope spinning. Over time, the effects of gravity will dominate, and the gyroscope will drift downward. For the brief period of time that the bottom of the wheel is being accelerated by gravity, it is moving down.
You can see this effect without a fancy motor/bearing system. Just wait until the gyroscope starts losing its rotational energy. Or, if you're impatient, just don't spin it quite as hard. It will slowly drop down, and eventually fall off the end of the stick.Source