Most modern airliners, particularly larger ones are powered by jet engines and for good reason. Jet engines are extremely efficient for high speed and constant thrust applications. A jet engine would be a poor choice for powering the family car for instance. Cars need to vary their speed frequently even on a long open road, it's unlikely that a constant speed could be maintained for very long and they never go particularly fast. Most aeroplanes on the other hand fly for most of the time at a constant speed and thus at a constant engine power setting.
A jet engine or more correctly, a gas turbine engine, is a practical application of Newton's third law. Newton discovered that for every action, there is an equal and opposite reaction. Sitting in a boat, you push against the riverbank. The boat moves away from the bank into the river. The force that you exerted on the bank is the action; the reaction is to move the boat in the opposite direction to the force.
A gas turbine engine works, in layman's terms anyway, by sucking in air using fan blades on the front of the engine. The air is then compressed using the screw compressor principle of forcing the air down a screw "thread" that continually decreases in width. The compressed air moves into the combustion chamber and then heat is then added by burning fuel in the combustion chamber. The heat causes the air to expand, which significantly increases the volume of the air. The increased volume of air has to go somewhere and exits at high speed through the jet pipe, back into the atmosphere. The force of the gas leaving the engine is what generates the thrust - Newton's third law. Some of the gas energy is used to turn the turbine in much the same way as a turbo charger in a car engine. This energy turns the central shaft of the engine to drive the fan and compressor stages at the front of the engine.
One of the reasons that jets work more efficiently at high altitudes is that the air is cold up there, very cold, around -50 degrees Celsius. Because of the temperature of the air, even though it is thinner than at sea level, the relative density is much greater.
The name jet engine came about due to the jet of gas being forced out of the engine. Going back to Newton's law, the jet efflux leaving the engine has a force or reaction, the opposite reaction is that the aircraft moves forward. It is often assumed that the jet efflux is pushing against the static air behind the engine. This however is a misconception; jet reaction is an internal phenomenon. It is simply the force of expelling the gas that generates forward thrust. Think of a garden hose with a high-pressure nozzle. I'm sure most people have switched on the tap and then had to chase the end of the hosepipe around the garden.
The downside of a jet engine is that they perform better at higher speeds and altitude but they become quite inefficient in the warm lower layers of the atmosphere. This can occur if the aircraft is kept low and slow due to other traffic. Bypass engines solve a lot of the lower level inefficiency problems. These engines compress all the air as it enters but then some of the air is ducted past the main part of the engine. Only the remaining portion goes to the combustion chamber. This has the effect of lengthening the tailpipe which improves efficiency and fuel consumption. It also has the highly desirable side-effect of making the engine quieter, appeases the airports neighbours.