The traditional definition of a rocket is a vehicle, missile or aircraft which obtains thrust by the reaction to the ejection of fast moving fluid from within a rocket engine.
There are many different types of rockets, and a comprehensive list can be found in spacecraft propulsion - they range in size from tiny models such as water rockets or small solid rockets that can be purchased at a hobby store, to the enormous Saturn V used for the Apollo program.
Most current rockets are chemically powered rockets (internal combustion engines) that emit an exhaust gas.
A chemical rocket engine can use solid propellant (as does the Space Shuttle's SRBs), liquid propellant (as does the Space shuttle main engine), or a hybrid mixture of both.
A chemical reaction is initiated between the fuel and the oxidizer in the combustion chamber, and the resultant hot gases accelerate out of a nozzle (or nozzles) at the rearward-facing end of the rocket.
The acceleration of these gases through the engine exerts force ("thrust") on the combustion chamber and nozzle, propelling the vehicle (in accordance with Newton's Third Law).