Shaun, some basic compressor info:
1. The compressor - this is the device that generates the airflow. There are several different kinds and from my perspective the most important thing is the 'noise' it makes. There are piston-types, diaphram types, and others that I'm not familiar with, but basically, they all pump air. There are big ones and small ones, but the smaller ones tend to be what most hobby people use (they are also less expensive).
2. The storage tank - some compressors don't have this, but my feeling is that it is a must. Lots of the 'hardware store' variety compressors that are used on construction sites usually have a compressor and tank complete as one unit. The tank basically stores the pressurized air so that the compressor does not have to work all the time. As mentioned above, you can just buy a 'tank' and connect it to your airbrush. The problem here is that you need to keep filling the tank somehow. The most common way is to use a 'compressor' (ie. #1 above!!), but you can also take the tank down to a gas station and fill it with the air hose that they usually have there (for filling car tires). Another cheap alternative is to get one of those small emergency air compressors meant for filling car tires on the side of the road. They usually connect to a cigarette lighter (12V) and will crank out lots of pressure, but at a fairly slow pace.
#3 A regulator (and gauge) - this is the device that controls how much air is released from the tank to your airbrush. If you were just to hook up the airbrush, it would spray at whatever pressure was in the tank!! (which could be very high, or very low.) Whatever the case, you want to be able to adjust this output pressure and the regulator is what does that. It usually has a gauge on it to tell you the air pressure it is set at. There is usually a knob of some sort that can be turned to adjust the pressure upwords or downwards.
#4 A moisture trap - this is also a must because as air gets pressurized and de-pressurized it tends to condense the water vapour into liquid. Sometimes this collects in the tank and can come thorugh the regulator. Water in the air line will contaminate the paint and cause all sorts of nasty things that you do not want. Depending the the type of compressor you have, there may also be 'oil' in the air line that the moisture trap will also remove.
So, that said here are the options:
a. get a small hobby compressor - this will likely not have a storage tank and will run constantly. I have never used one like this so I'm not really sure how they work. Likely a $50-100 option.
b. get a tank and fill it at a gas station - this is probably the cheapest, but depending on how much you use your airbrush, you'll need to keep filling the tank. You'll also want a regulator to control the air volume output. Likely a <$50 option.
c. get a tank and a small hobby/emergency compressor - this way, you can fill the tank with the compressor whenever it gets low on air. You'll need a regulator for this too. Likely a $50-100 option
d. buy a purpose built compressor/tank combination - likely the most expensive, but it will have everything you need with the possible exception of a moisture trap. There's lots of options here and the range of $ is pretty wide. You can get a nice small compressor from Sears or another hardware-type place for $100-200. Generally these compressors aren't really that 'quiet'. If you need 'quiet' then you have to look at other options, likely specific hobby compressors.
e. get a CO2 tank and have it filled at a welding gas supplier. These tanks are the most expensive, but do not have a compressor. They are filled with CO2 gas by a supplier to very high pressures. They usually come with a regulator to turn the pressure down. The advantage of these is that the CO2 tanks last a long time and the air does not need a moisture trap. There is not compressor, so therefore no noise. Depending on how often you use compressed air, you'll likely have to have it filled once or twice per year. You'll have to buy the tank to begin with (rather expensive), but then you just pay for the refills (very cheap).
That's alot of info, so I'll let you digest and invite others to also offer advise.
M.