The basics are that you want to be able to fix a flat while you're miles from home. For that you'll need:
Tire levers (the blue things) used to pry the tire off the rim so you can get to the tube inside, and patches (little rubber squares or circles, plus some glue) to fix a leak in a tube.
Or an extra tube instead of the patches.
Small pump or CO2 inflator to fill the tire back up again:
If your bike doesn't have quick releases (little levers) on the axles, you'll need a wrench to take them off. That's usually a 15mm and will probably be cheaper from a hardware store.
Those things plus a little bag to put them in under the seat will usually get you home in one piece.
Last piece of advice: Take a look at that link bendembroski posted, take your back wheel off and turn the tire around for practice so you'll know how to do it if you have to on the side of the road some day. Then have a beer.