I'll try and put it in laymans terms for the OP, and hopefully explain whats going on a bit. Not that the previous posters are incorrect in any way, i'm just a little concerned they might be pitching at a level above the original poster. Sorry if it is a little too "noddy level" for the OP.
First of all, a Jet and a Car Engine work in the same way. They have to take in air ("Suck"), compress it ("Squeeze") burn fuel in the compressed air ("Bang") and finally exhaust the used gas ("Blow"). Whereas in a car, all the above operations (suck, squeeze, bang, blow) take place inside a piston, in a Jet, they are split into separate parts of the engine. These parts are (from front to back) the Fan, or N1 section (Suck), the Compressor (Squeeze), the combustion chamber (Bang) and the turbine (Blow). The Turbine is split into (usually) two sections, which are connected by a driveshaft to the Fan and compressor sections, and use the exhause flow from the combustor to drive them.
So, what has all this got to do with the price of fish, or starting engines? well, in a car, you turn the key, which fires up the starter motor, turning the crankshaft, which turns the pistons, which suck, squeeze, bang, blow, quicker, and quicker, and eventually the engine reaches a speed which disconnects the starter motor, and the engine burbles away quite merrily.
A jet engine works in exactly the same way. The pilot will press a button (No keys to forget!) which will start the compressor section turning, using either a motor (like a big version of a starter motor) or high pressure air. At a particular speed, the pilot will move a lever, which will allow fuel into the combustion chamber. The igniters (like a car spark plug) will be switched on as well. This will cause the fuel to burn (not explode!), causing more high temperature, high speed gas to turn the turbine, causing the compressor to turn faster, and in time, the fan too (remember it was not connected to anything other than its part of the turbine?). When the engine reaches a certain speed, the starter will disconnect, and the engine will keep accelerating until it reaches a point where it hums merrily away. This is why the fan can appear to be standing still, turning the wrong way, or spinning for a while before the engine starts.
As has been previously explained, each part of the engine is very well balanced, and often quite light, in fact it is often possible to turn a fan blade by hand (although you wont be able to give it a push start!)
Hopefully that has answered your questions. As I said, all the other contributors have answered the question as well, but I just get a bit of an impression that the above might answer things a bit more fully. And pay no notice to Kevlarcarl, god knows I have difficulty enough telling engineers what is going on with my aeroplane, from outside the cockpit I can imagine some of the things that happen must at best be a bit baffling, at worst downright scary!
TTR