When referring to a "rocket" we usually mean a large object that uses chemical reactions to propel itself at high acceleration to reach very high velocities. Rockets are used primarily to send cargo, people, satellites, and rovers outside the earths atmosphere to orbit or entirely other planets, although some "rockets" you could say are strapped to other vehicles like cars or planes, but we care mostly about the space kind. The reason rockets function in the way they do was given an explanation in the form of Newtons Laws of Motion.
Newtons 1st. Law - An object in motion will stay in motion unless acted upon by an outside force. A rocket must continuously apply a force upward to counteract the forces of gravity.
Newtons 2nd. Law - This law just states that the force of an object is equal to its mass times its acceleration (F=ma). Pairing this with the first law we can quantify the amount of thrust needed to accelerate the rocket upwards towards space.
Newtons 3rd. Law - For every force there is an equal and opposite reactionary force. This is what moves the rocket, when it puts out thrust away from it, it is creating a force in that direction, thanks to the third law this puts the same force upward on the rocket.

Although the overarching governing laws for the rocket and its motion, Newtons laws are not the only thing going on here. The way rockets work is also explained by thermodynamics, energies, and aerodynamics. The first law of thermodynamics states that energy can not be created nor destroyed, it can only change form from one type to another, and this is precisely what happens inside the rocket when it take the energy inside the fuel and begins combusting it converting that chemical energy to kinetic, combustion, mechanical, and other kinds of energy to create thrust.

Benson, Tom. Rocket Principles. [1]