Let's walk through the stages of a jet engine. The first part is the compressor. The compressor is generally made up of multiple stages and each stage is designed to compress the air a little bit more (increase the pressure). This is done because high pressure air is high density air (and hot air), and the combustion that will occur later is more efficient at high pressure and density. The compressor stages themselves are a big wheel of blades, and each blade looks like a little wing. As the wheel spins, the wing generates some lift and pushes the air deeper into the engine, compressing it as it goes. Compressing the air requires energy and the compressor is driven by a shaft connecting it to the turbine.
The second part is the combustion chamber. The air entering here can be up to 40 atmospheres pressure. Fuel is injected and burns, releasing the chemical energy. This chemical energy release increases the temperature (a lot, but one must be careful that it isn't too much or your turbine will melt).
The last stage is the turbine. There is hot, high pressure air in the combustion chamber and it moves rapidly into the turbine. Like the compressor, the turbine will generally have multiple stages. Again, if you look at each individual blade in a stage, it looks like a little wing. These wings generate lift as the air moves past them, which in turn spins the turbine. The air moves past because it is high pressure in the combustion chamber and low pressure outside the engine. This spinning of the turbine drives the shaft that spins the compressor. Once the cycle gets started, it is self-maintaining (at least, until you run out of fuel). Of course, the turbine cannot extract too much energy or there wouldn't be anything left for thrust. Unless of course you are using a turboshaft or turboprop, where you want the turbine to extract all the energy possible and what isn't used to spin the compressor is used to drive the shaft or propeller.
There's obviously a start-up problem. If the turbine spins the compressor and the combustion needs the compressor running and the turbine needs the combustion, something has to spin it to start. This can be done with giant blowers that force a large amount of air through the compressor so it begins to spin. Or it can be done with another power unit on board (APU) which is designed to get the compressor started so the combustion can take place and the turbine can drive the compressor.
That is jet propulsion in a nutshell. There is way, way more involved when you get into the nitty-gritty details, but that's an overview of what each step does and how it connects to the others.