The contents of a cyclists seat bag can, literally, mean the difference between a great ride and one that you would soon forget. Sure, we all would like to have every ride be trouble free, but let's face facts. Stuff happens when you are on the bike and if Murphy is along for the ride, it seems like things head south at the worst possible time.
Well, first things first. You need to have a good pump. Not CO2 cartridges, a good pump. CO2 is fast and easy, but if you somehow screw it up or your spare tube has a leak or you get multiple flats you may be walking. So get a good pump and know how to use it. OK. If you really want to use CO2, fine. But, bring a pump along just in case the CO2 fails. It will, trust me.
Inside your seat pack you should have at least the following essentials. Two, or even better, three tire levers for prying off the tube. Tire levers seems to break when least expected so make sure you have at least two. A spare tube is critical. Wrap it in plastic or keep it in the cardboard box to protect it from getting punctured in your seat bag. If you ride in areas that are known to be flat-prone either carry a second tube or better yet, a patch kit. If your patch kit uses glue, make sure the glue has not dried out. I carry both glue and glueless patches just in case.
One last essential is some form of tire boot. If your tire gets a cut that would allow the tube to poke through, you need to put something inside the tire to prevent that. I use Tyvek, the strong, paper-like fabric which is used by Fed Ex and the USPS for their mail envelopes. In an emergency, a dollar bill or energy bar wrapper will suffice.
Some non-essential, but very handy items include a spoke wrench which can come in very handy if a spoke lets go. Of course, that means you have to know how to use it. It's pretty simple. Loosen the two spokes on the opposite side of the rim which are on either side of the broken one. Use small turns until the wheel clears the brakes. A small screw driver is handy for making on-the-road derailuer adjustments.
A set of allen wrenches, 3mm-6mm, are handy for adjusting seat height, stem, and other allen key fittings on your bike. Also, a chain tool is useful if you are the type who seems to break things. I carry a $20 bill and a credit card in my seat pack. Money can't buy everything, but in a really tough jam it just might help.
Obviously, your mileage may vary(YMMV), but there's a good start. Enjoy the bike, but if Murphy shows up for the ride. Watch out!
Bruce
Active Expert: Bruce Hildenbrand
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Jason
I think you should have a patch kit regardless of whether you have a spare inner tube...for that second flat. This is also why I pretty much turn my nose up on CO2 inflaters. What happens when you get nailed a second time?
Some disposable examination gloves are a great convenience, too, to keep your hands from getting totally roached, since it always seems to be the rear tire that has to be fixed.
Carry a couple of zip ties. Not only do they weigh nothing, that can make the difference between walking and riding should a fastener give way.
Some people swear by the "spoke in a can", that allows you to actually replace a spoke. This is very important if you're running a low spoke count wheel.
The one last thing you could consider but is not essential is a cheap Leatherman-like folding pliers. It can really help with certain repairs and with pulling pieces of wire and glass out of your tire. Put your tools in a plastic bag to reduce corrosion from moisture.
John Sully
I like carrying one of those screwy littly Ritchey multitools for small adjustments and fixing broken links. Carry a pump, the longer the better. If you have a road bike, carry the largest pump you can put between your bottom bracket and the top tube, the shorter ones just don't work as well. On a mountain bike shorter pumps are OK, since you don't have to get to as high a pressure.
I once went on a ride where I got three flats. After that one, I got slightly heavier kevlar belted tires. Flat problems went away, but I still carry a spare tube and a patch kit. Doing that saved me from walking 15 miles on the three flat trip. Oh yeah, I usually wrap the spare tube in an old sock, probably remnants from the day when any self respecting cyclist rode on sew ups. BTW if people still ride sew ups, carry a spare and a patch kit and make sure you have a needle and thread 
.
BTW, with practice you can usually get a tire off and on without having to use tire levers (tire irons in the old days, or maybe aluminums). Usually you need at least one to get it off, but I usually carry a whole set of 3. It is usually better to work the tire onto the rim with your hands with the new or patched tube though.
I got a great comment from my friend Larry. For a whole host of good reasons everyone should carry some form of ID when they ride. It may seem reasonable to carry your ID in your seat pack, but if you get seperated from your bike in the event of a crash (hint: they won't bring your bike in the ambulance with you to the hospital). Larry recommends RoadID which is a dog tag which you wear that has all your pertinent information. Check it out.
Bruce