A spinning wheel has angular momentum, which gives it a natural tendency to keep spinning at the same speed in the same direction. When you tilt the gyroscope, you change its angular momentum. The spinning wheel resists this change, pushing back and causing you to spin in the chair.
The natural tendency of spinning objects to resist changes in direction can be useful.
Have you ever wondered why bicycles are easy to balance when they’re rolling, but tend to fall over when they’re not? The bicycle wheels are gyroscopes that naturally resist tilting—at least while they’re turning.
When a quarterback throws a football, putting a spin on the ball helps keep it pointing forward as it travels through the air.
Ships, aircraft, and spacecraft use gyroscopes in navigation equipment to help stay on course. Mounted in a frictionless cage, a gyroscope maintains its orientation no matter how the craft moves. Gyroscopes in the Hubble Space Telescope help aim it by keeping constant track of which way is up.