The best ways to unclog a toilet are to use a plunger or an auger to break up the clog and remove it. If a plunger or auger is unavailable, it is possible to break up a clog with really hot water and dish soap.
A toilet clogs when toilet paper and waste pile up and become too large to flow through the plumbing system. Some toilet clogs will just prevent the toilet from flushing, while others will cause the water to flow up and out of the toilet. Using a flexible plunger is important, as stiff, hard plungers aren't as effective at creating an air-tight seal between the toilet and the plunger.
Using an up-and-down motion, plunge the toilet clog until the water begins to flush. If the water won't flush down, an auger may be necessary to manually remove the clog. An auger can fit through the toilet hole and either break up the clog or grab it so that it can be pulled back up through the toilet hole to release the clog.
For the times when a plunger isn't available, really hot water and dish soap can help to break up a clog. Dump a gallon of really hot water into the toilet along with a few drops of dish soap. Let it sit for a few minutes and then try to flush the toilet. If it still won't flush, then a plunger or a plumber may be necessary.