Toilets are surprisingly simple appliances. As such, they're generally fairly simple to fix. A wobbly toilet is no exception. Only a few things can set your toilet to trembling, and for most of them, diagnosing and repairing the problem is a cinch.
It's possible, though unlikely, that your toilet's closet bolts--the bolts that hold it in place--have come loose, and this is causing the wobble. Remove the caps covering the closet bolts and check them for tightness using just your fingers. If you can't loosen or tighten the bolts by hand, this isn't the problem. If you can, use a wrench to tighten bolts. You don't want to do your Hercules impression here--if you overtighten the bolts, they crack the porcelain and you end up needing a new toilet. Once the bolts are tightened, check the toilet. If it is stable, great, you're done. If not, you need to take a look underneath.
Remove the Toilet
Begin by closing the toilet shutoff valve. Next, remove the tank cover and flush the toilet, holding the handle down until the tank is completely drained. (If the tank starts to refill, you need to turn the shutoff valve a little harder). Use old towels to sop up any water that remains in the tank. Remove the nuts from the closet bolts, and carefully lift the toilet up and set it down on some newspapers or a collapsed cardboard box. Most toilets aren't insanely heavy, but they are awkward, and they do break when dropped, so consider using a helper when you do this. Finally, use a rag to plug the drainpipe.
Examine the Closet Flange
The flange is the cast iron ring to which the toilet is bolted. If it is bent or broken, your toilet wobbles and, eventually, leaks. In most cases, the flange can be repaired, which is described in the following section. If the flange appears to be intact, you should be able to fix the wobbling by replacing the existing wax ring with two new ones.
Use a putty knife to remove the old wax ring from the flange and the bottom of the toilet. Replace the old closet bolts with new ones, and place your two new wax rings on the flange, one over the other. If the wax rings are not at room temperature, give them some time to warm up--this makes them easier to work with. You can also brace the closet bolts against the wax rings to keep them upright. Set the toilet back into place, making sure the closet bolts are aligned, and press it firmly down so that it is well-seated on the wax rings. Put a plastic washer, then a metal washer, over the closet bolts, and then tighten the nuts, being careful not to overtighten. Use a hacksaw to trim the bolts down to the nut, and replace the plastic caps.
Repair the Flange
If the flange is broken or bent on one side (in the vicinity of one of the bolts), you can fix it with a flange repair plate (aka flange anchor). If both sides are broken, you need a new flange, a tricky job best handled by a licensed plumber. The flange repair plate fits under the bent or broken part of the existing flange, and includes a slot for the closet bolt.
Actually, once you see the repair plate, the proper way to install it should be obvious. The only potentially tricky part is that you might have to chip away the tile under the existing flange to make room for the repair plate. Once the repair plate is in place, install a new wax ring (or rings--using two requires a bit more effort, but makes for a better fit) and new closet bolts as described in the previous section.