What typically causes a toilet to leak?
If it's the toilet itself, you'll see water on the floor around the toilet -- water will leak from the union of the tank (backrest) and the bowl. Make sure that the tank is attached tightly to the bowl (bolts are under the joint) and that it cannot rock. 

If it's the seal or the plumbing beneath it, you will see water flowing from underneath. You'll need to replace that drywall under the toilet and get it all dried out anyway to prevent mold, so you might as well pull it down. 

Wax seal rings are cheap, and unless the water is coming from the tank or you see a cracked pipe in the floor, are probably what you're going to need to replace. Just go to your local home improvement store and get a "Wax Ring Kit" and you should be good to go.