How Reaction Engines Work in Rockets

The simplest rocket motor, and the first, is the black powder engine. Developed in China around 1000 AD, the black powder rocket involves little more than a tube filled with gunpowder and a clay nozzle to speed up the exhaust gasses. Black powder is a mixture of sodium nitrate, sulfur, and powdered charcoal. These are ground together wet, then the paste is packed into a tube, and a plug of clay with a hole in it is inserted into the end of the tube. Once dry, the fuel and oxidizer in the black powder is ignited. Oxygen in the chemical structure of the sodium nitrate combines with sulfur and carbon releasing energy and forming carbon and sulfur dioxides. The hot gasses are expelled through the hole in the clay nozzle and the jet of gas imparts a reaction force on the tube which forms the rocket. This is still how simple fireworks rockets operate.
Liquid fueled rockets were developed in the 1920s by Robert Goddard and various scientists including Werner Von Braun in Germany. In a liquid fueled rocket such as the V-2, the liquid fuel, a mixture of water and alcohol, combines with the oxidizer, red fuming nitric acid, in a spherical combustion chmber. These two components were chosen because they are hypergallic, meaning they ignite on contact, making an ignition system unnecessary. The fuel and oxidizer burn in the combustion chamber. Heat causes the product gasses to expand, and they are ejected through a bell-shaped nozzle to form the exhaust stream. As the exhaust gasses accellerate out of the nozzle a force is generated on the rocket motor which pushes the rocket upwards.
While they are simple in principle, rocket motors can be hard to make because of the extreme pressures and temperatures they generate.
Perhaps the most difficult part of rocket design is to make the rocket stable in flight. Early black powder rockets used a simple stick to stablize their flight. The weight of the stick brought the center of gravity of the rocket behind the center of thrust, so the resulting flight was stabilized in direction. More modern rockets that operate in the atmosphere use aerodynamic fins to counter any turning forces on the rocket body. Most missiles use some form of control on the fins or vectoring of the thrust of the engine to steer and align them with their targets. In space, attitude control jets, small rocket motors, are used to orient the vehicle in direction.
There are many developments in rocketry, such as higher thrust engines and better control systems, but they are all based upon the basic idea of a jet of gas providing thrust.