Modern solid fuel charges are usually molded from synthetic butyl rubber, with potassium permanganate as an oxidizer.
The fuel is cast in different forms for different purposes. Slow, long burning rockets are solid. Faster-burning motors are cast with a single-star-cross-sectioned hole down the center of the motor. Faster-burning motors, such as those used in hand-launched missiles, have mutliple holes that rapidly grow in burning area.
One of the many drawbacks to solid rocket systems is that once they've been ignited, they can't be turned off. Some solid fuel motors do have an auxiliary valve, usually at the front of the motor, that allows their thrust to be selectively reduced in a controlled way by venting reaction gas.
Solid fuel motors aren't commonly used in satellites; satellites need rockets that can be fired multiple times (or pulsed.) However, see hybrid rockets, which use solid fuel and a liquid or gaseous oxidizer that allows them to be throttled.
See also: Spacecraft propulsion