I was doing research about jet engines, and they seem really difficult to fully understand. So, can anyone explain it in a simple way?

An afterburner glows on an F-15 Eagle engine, source: Wikimedia.
How do jet engines work?
I was doing research about jet engines, and they seem really difficult to fully understand. So, can anyone explain it in a simple way?

An afterburner glows on an F-15 Eagle engine, source: Wikimedia.
How do jet engines work?

Users reaching this question: How does a jet engine work? (explanation for non-aviation related people) are now redirected here. This illustrative answer is in the spirit of the initial question, easy to read for a non-aviation person without sacrificing accuracy.
To create a reaction, throw something
The physical principle at work has been studied by Isaac Newton and is known as principle of reaction (or the more mouthful Newton's third law of motion).
The reaction movement is the response to another movement. In many cases, including the jet engine, this is done by moving something heavy, at the highest possible velocity.
While reaction seems magical, it's something experienced non-stop in everyday life, without giving attention to it. We can easily imagine the result of such experience:

A note here about the velocity:
In the boat experiment, if we want to throw the rock without moving the boat, instinctively we'll move it very very slowly until it is above water, then let it drop into the water without pushing, and it'll work.
We also know what happens when a big gun throw a 500 g shell at the speed of sound: There is a huge reaction of the gun, in spite the gun is heavier than our boat and friction on the ground should prevent it to move.
Feel reaction intuitively:
When we push on the rock, we are actually using the rock as a fulcrum, and creating the reaction this way, we push on the rock, we move the boat. However to continue generating thrust, we need to push on the rock stronger and stronger, because the rock itself is accelerating due to the force we apply to it. If we just move our hands at the speed of the rock, we are not pushing, the motion of the rock is unchanged due to second law of motion, there is no impulse, thus no change in momentum, and the boat just starts being slowed down due to drag (from water), and the rock too (drag from air).
Actually the force effects we create by pushing (accelerating) the rock are split between the rock and the boat according to the ratio of the masses. The largest mass gets the smallest change.
Reaction from throwing air
Throwing stones is obviously not practical for an engine. But we can throw air, like in a balloon, again something very familiar:
This principle of ejecting air to create a reaction has been known since Antiquity with the aeolipile. It was an early steam engine. Air was ejected through tangential openings at the end of pipes.
A jet engine works the same way, throwing air overboard, in large quantity and at very high velocity. Compared to the boat experiment, we have replaced:
The stone by air. As air is much less dense than rock we need to concentrate a lot of air to get the same effects.
Hand muscles by fuel combustion. Combustion delivers high velocity gases by effect of gas expansion at hot temperature. The more fuel, the higher the gases velocity.
Burning a lot of fuel requires a lot of air, we have already a need of air in large quantity, so this is not a problem. However air won't come in large quantity into the engine, at some point air already present in the engine prevents more air to enter. In flight, ram air pressure can increase the quantity of air in the engine, but this isn't enough.
The solution is to use a compressor to fight the pressure of air already in the engine, and push, on large airplanes, about one ton of air per second in it. It's only 50 times more than our stone, but the exhaust velocity is definitely not the same: About 1,000 km/h.
How do we collect a lot of air, and eject it at high velocity?
We have all the ingredients to make a jet engine, which is composed of three sections (to simplify, let's look at an early turbojet engine, refer to this question for common turbofan):
Air enters in the engine from forward and is compressed by a pump (named the compressor) to increase the quantity available.
Compressed air is mixed with fuel and lit. Air is required for fuel to burn (oxygen). During the combustion, the mix achieves a very high temperature and expands under the effect of heat, exactly like water vapor expands in a pressure cooker or an aeolipile.
Before being ejected as small part of the hot gases energy is used to turn a turbine (like wind turns a wind turbine). This rotating motion is transferred to the compressor we saw at the beginning. To start the compressor at the beginning a starter is used, like on a car.
Hot gases are ejected rearwards this creates the forwards reaction of the engine.
The components of a jet engine are therefore:
There are three basic parts to a gas turbine, a compressor, a combustion chamber and a turbine. Air is compressed by the compressor, heated by the combustion chamber and the heated air powers the turbine. The turbine in turn powers the compressor.
The key is that the heating in the combustion chamber causes the air to expand. This means that the work that can be done by gasses (air plus combustion products) exiting the pressurised combustion chamber is greater than the work needed to force the incoming air (and fuel) into the combustion chamber.
The available energy in the exhaust can be used in various ways. The simplest is the turbojet, the stream of hot gasses coming out the back of the engine directly provides forward propulsion.
Alternatively we can capture more of the energy from the exhaust with a turbine and use it to drive a fan resulting in a turbofan.
Or we can design our turbine to capture most of the usable energy in the exhaust and direct it to a shaft which can be used to power all sorts of things.
A jet engine is an overcomplicated ramjet with extra turbines to let it work at lower speeds.
A ram jet works by igniting compressed incoming by mixing it with fuel and providing a spark.