What are the methods of rocket propulsion?

Answer 1

All methods of rocket propulsion are based on the exchange of momentum between the rocket and the fuel. This basically means throwing something behind the rocket at high speed. In essence, throwing a ball away from you is the same principle, you will experience a small force as the ball is thrown, a rocket does this on a huge scale. Many things can be used the achieve this affect

Chemical Rockets - Ignite a combustible fuel under pressure and use the expelled water and other by products to achieve a thrust. The escaping water is travelling at a huge speed so the rocket moves. Contrary to popular belief, fire does not = thrust.

Ion Propulsion - Using accelerated ions by using a system much like a mini LHC to provide a very small thrust, generally only used by small, unmanned craft due to practical limitations of this technology.

Nuclear Propulsion - The most effective form of propulsion ever conceived, drops tiny nuclear devices behind the craft and detonates them, using the expelled matter from the explosion to propel the craft. These craft were developed by the USA in the 70s but are now illegal under the partial test ban treaty. (See Project Orion)

Other forms - Accelerating rogue dust particles in space to achieve thrust (not very effective), using antimatter particles and using them as annihilation charges in a similar system to nuclear propulsion (the holy grail of space travel). Multiple theoretical engines exist, worth looking around on the internet.

Answer 2

An important principle is Newton's Third Law (or alternately, Conservation of Momentum, which is more or less equivalent). In this case, if the burning gases go out of the rocket at a high speed, the gases and the rocket push each other in opposite directions. - An alternative, but equivalent, explanation is that total momentum is conserved.