How to Start a Jet Engine
Inside some jet engines there is a powerful electric motor that starts the engine by spinning the compressor up to speed, at which point the fuel flow and the ignition spark are turned on, and the resulting combustion then forces the turbine to spin. The turbine's power output is sent to the compressor stage to spin it, and the electric start motor can then be turned off.
On other jet engines, a small gas turbine is used to blow hot combustion exhaust through the main turbine, which spins the main compressor up to speed, then the fuel and spark come on, etc. The start turbine is then shut down once the main engine starts.
Note that whenever the engine is running, the compressor stage at the entrance to the engine is actively sucking in tremendous quantities of outside air and jamming that air into the combustors in the center of the engine, where it is burned with fuel. The power of the exhaust from the combustors then spins the turbine at the outlet end of the engine, and that power is fed back to the compressor on a spinning shaft to keep it sucking in more air.
Some military jet engines can be started by connecting a huge blower to the engine inlet and forcing compressed air through the engine, to spin the turbine and get the compressor going. This is called an air start. Some military jets can also be started by burning a solid-fuel cartridge made of black powder and shooting the hot exhaust into the turbine stage of the engine to spin it, and thereby spin the compressor stage up to speed and get things going.
Other military jets are started by a start cart which contains a powerful gasoline engine that spins a shaft geared onto the main shaft of the jet engine. Once the jet is up and running, the start shaft is disconnected from the jet and the start cart is rolled out of the way.