A satellite stays in orbit because its velocity is sufficient to counteract the Earth's gravitational pull. All objects near Earth are influenced by its gravitational pull and tend to fall towards the planet. However, if a satellite is moving fast enough parallel to the Earth's surface, instead of falling down and striking the planet, it will effectively fall past it instead, maintaining a circular orbit.
When a pitcher throws a ball horizontally, it may travel a great distance, but eventually gravity will pull it down to the ground. If the pitcher could throw it hard enough, however, the curvature of the Earth would cause the ground to fall away from the ball at the same rate at which gravity pulls the ball downward. This would create a stable orbit. Thrown even harder, the ball would travel faster than the force of gravity and fly away from the planet.
Matching these forces is a tricky business. Satellites must orbit hundreds of miles above the Earth to maintain enough velocity to counteract the Earth's pull. Satellites must also orbit far enough out that the Earth's atmosphere cannot provide drag and slow them down. Most orbits eventually decay, however, requiring satellites and spacecraft to use propulsion systems to maintain enough velocity to remain in orbit.