The planets all formed from this spinning disk-shaped cloud, and continued this rotating course around the Sun after they were formed. The gravity of the Sun keeps the planets in their orbits. They stay in their orbits because there is no other force in the Solar System which can stop them.
The initial speed of the satellite maintained as it detaches from the launch vehicle is enough to keep a satellite on orbit for hundreds of years. A satellite maintains its orbit by balancing two factors: its velocity (the speed it takes to travel in a straight line) and the gravitational pull that Earth has on it.