Rocket propulsion is one of the applications of Newton's third law. If you push something in one direction that thing will push you with equal force in opposite direction. Simplest example of rocket propulsion is 'a balloon'. You inflate a balloon and then release it, high pressure air while escaping through the hole throws high momentum air to the surrounding (in other words pushes the surrounding air). As a reaction according to Newton's third law, the surrounding air gives equal amount of 'push' to the balloon in opposite direction, and your simple rocket (balloon) flies.

The real rocket rocket works on same principle but it does need to generate that 'high pressure air' like balloon little differently. Worse, it has to generate that in outer space where there is no atmosphere. Therefore the rocket carries fuel (liquid or solid) and oxidizer in containers inside its body. These (fuel and oxidizers) are burnt in a combustion chamber and generate high pressure and high temperature gases that rush out of the nozzle of the rocket (just like high pressure air rushed out of balloon). Rest is same as balloon.   

see

https://www.grc.nasa.gov/www/k-12/airplane/rocket.html

http://web.mit.edu/16.00/www/aec/rocket.html

"Rocket Propulsion Elements" by George P. Sutton is one of the best book in this regard soft copy is available here 

http://www.ewp.rpi.edu/hartford/~ernesto/S2013/EP/MaterialsforStudents/Lee/Sutton-Biblarz-Rocket_Propulsion_Elements.pdf

Rocket propulsion is basically non-air breathing(do not use Oxygen from Air for combustion)  jet propulsion which are mainly of following type:

1) Chemical Rocket propulsion (Mostly used in Spacecraft escaping earth gravity)

2)Electric propulsion ( Mostly used in spacecraft in the vacuum)

3) Nuclear Propulsion