Jet (Gas-Turbine) engines are usually thought of as thrust producing engines and not engines that drive rotors or propellers. Jet engines are used for both thrust production and for drive power. If you have ever heard of "turboshaft" or "turboprop" engines, they are basically jet engines that do not produce thrust. They take the power from the production of thrust and turn it into mechanical power to drive rotors or propellers. The way the turbine blades are shaped determines whether they change thrust into great amounts of power to drive a shaft, or if they allow most of the thrust to be used as propulsion and take little drive power away to just drive the compressor section of the engine.
Helicopters (In almost all cases where jets are used) use turboshaft jet engines to power their rotors. The UH-1 Huey, the Bell Jet Ranger, and the AH-64 Apache all derive their power from gas-turbine engines. Some older or smaller helicopters use "reciprocating" (Piston) engines for their power source, but most of the helicopters in use today use gas-turbine engines. They are light, very powerful and economical. The best part is that they are very reliable as well. Failure rates for gas-turbine engines are very low because there are not as many internal moving parts as there are in a reciprocating engine.
Let's look at how a jet engine works. A jet engine works on four very simple principles: "Suck, Squeeze, Burn and Blow". (Stop laughing now...this is serious stuff) The picture posted here shows a simple gas-turbine engine cross section. In the front of the engine is the compressor section which "Sucks" in air and "Squeezes" it to make it more dense and better for combustion. Air is brought into the compressor by the turning compressor blades that are shaped like little airfoils. It works like a big fan to move air into the engine. In between the moving rows of compressor blades are stationary blade sets called "stators". The stators change the direction of the airflow and help in the compression process. As you can see by the picture, the area that the air can occupy gets smaller as the air travels through the compressor. The air then goes through the diffuser section which transports the air neatly into the combustion chambers, which are in the combustion section. There the air is mixed with fuel and is ignited to create a powerful reaction. (The "Burn" part)
The explosive burned fuel and air mixture then travels into the turbine section where the force is turned into a combination of drive power and thrust (or exhaust). If the force is converted mainly into drive power to drive a transmission, as in most helicopters, then it is referred to as a turboshaft engine. If the force is only converted to enough power to drive the compressor, and the rest is used as thrust, then it is considered to be a turbojet (or thrust producing) engine. (That is the "Blow" part). It is a really basic design, which can get a lot more complicated than this example, but for theory sake, this is an adequate representation of what happens.
The power is transferred from the engine using a main gearbox which changes the power from the engine and sends it to the transmission. In the transmission RPM is reduced from thousands of RPM to hundreds of RPM. By doing this the torque is increased and the rotation is slowed to an acceptable level for the rotor system. The transmission drives the mast which gives direct rotation to the rotors. Often another shaft will come out of the transmission to directly drive the tailrotor as well.
An accessory gearbox mounted on the engine draws little engine power to drive things like the oil pump, the generator and the fuel control for the engine itself.
Some current helicopters still use piston engines as their power source. Air cooled horizontally opposed engines are the most common aviation engines in use. Often called "Boxer" engines, they got the nickname from the way they "punch" outward. They are very smooth, well balanced engines, and their air cooled nature make them extremely versatile for use in all weather conditions. The main difference to the pilot is the need to regulate the throttle, where gas turbine engines are automatically controlled to maintain a certain RPM range.
Piston engines used in helicopters operate on the same principles as those in most cars. The parts are generally the same as well. A piston travels the length of the cylinder, compressing a fuel / air mixture, burning that mixture to provide power, and exhausting the emissions out of a tailpipe. Intake and Exhaust Valves open and close to allow the fuel / air mixture in and let exhaust out. A Spark Plug ignites the mixture and drives the piston downward. A Connecting Rod connects the piston to the Crank Shaft and the crank shaft drives a transmission to provide power to the rotors.