What is the purpose of a tank (for a compressor)?

Lady Jemima,
I would recommend an air tank on "any" compressor that you purchase for whatever "extended" spraying purposes.

An air tank serves two major functions -
One, as stated above, to remove any unwanted pulsation that can travel through the air line and hence show-up in your spraying results with the airbrush (or in any air tool).
Two, as a reservoir in which your air tool (air brush) uses the air from instead of directly from the compressor. Thus giving the compressor time to shut off and cool down which in turn extends the life of the compressor to some degree. The bigger the tank the better, but even the small size of the tank on the hobby compressor is benifical.

Just look at any compressor at any automotive garage. They all have large tanks.

It's worth the extra $40.00 or so in the long run... 

LJ - I recommend the tank, because unless the compressor you get is one of those $400+ dollar ones, it will have some amount of noise... a tank will only necessitate it being refilled in intervals vs. almost constantly on..

Also as Bill mentioned it will also dampen out any pulsation the compressor might cause...

Depending on where you live in the country a moisture trap is also a very good thing... unless you live in the desert it is something that will help a lot.  You can get one at the hardware store (Lowes/Home Depot/Wal Mart even) for about $11 to $20...   You don't want a big water spatter on freshly sprayed paint!!

Good luck!!!

Tanks are a good thing to have, especially with small diaprahm compressors. They eliminate any pulsation that the compressor might introduce to the air flow. If the tank is large enough the compressor should shut off after it fills the tank, and only come on when the pressure drops below a certain amount which is a lot quieter! If you can afford it, get the tank, but make sure you also get an moisture trap for the airhose. You want the trap to be as close to the airbrush as possible. You should get the trap regardless of the compressor you get.