The motion of a rocket is an application of Newton's third law of motion and law of conservation of linear momentum.
A rocket is a projectile that carries the rocket fuel and the oxidiser, which supplies the oxygen needed for combustion. Liquid hydrogen, liquid paraffin etc., are used as rocket fuels and hydrogen peroxide, liquid oxygen etc., are used as oxidisers. The fuel-oxidiser combination in a rocket is called the propellant.
The simplest form of a rocket consists of a combustion chamber in which a solid or liquid propellant is burnt. There is a nozzle at its tail through which the gaseous products of combustion can escape. The rocket forces a jet of hot gases downwards through the nozzle. This is the action. The jet of gases exerts an equal force on the rocket, pushing it forward. This is the reaction. This force gives the rocket a forward acceleration.
The operation of a rocket illustrates the conservation of momentum. Just before launching, the momentum of the rocket is zero. When the rocket is fired, it forces a jet of hot gases with a high velocity through the nozzle. The jet of gases acquires a momentum downwards. Hence, the rocket acquires a momentum of equal magnitude in opposite direction. Thus the rocket moves upwards.
With a single stage rocket it is not possible to attain very high speed and hence multistage rockets are designed. In multistage rockets when the fuel of the first stage gets exhausted, the rocket casing is detached and dropped off and the second stage is ignited.