Many problem toilets can be cleared using a plunger, but occasionally you can get a super clog. The only solution then is to break out the snake.
The majority of clogged toilets can be cleared by using the old pumping action of a plunger. Every once in awhile, there is going to be a clog that resists. The next step is to apply a snake or auger. The snake is a tightly coiled piece of spring steel with an auger tip on one end and a cranking handle on the other. There are two basic types. One is called a drain auger. The other is a closet auger.
The closet auger is shorter and has a stiffer coil. It is actually designed for use in the toilet. Many closet augers come with a protective, rubberized collar over the insertion part of the tool. This prevents scratching of the enamel of the toilet bowl when inserting it. The drain auger can come in many different sizes and also can be powered. On some, the crank handle can be attached to an electric drill to provide the turning power. Other models, designed for the use of professional plumbers, are self powered and can be quite expensive.
The coil of the snake is inserted into the mouth of the toilet drain. Slowly insert the coil into the drain. It is important to move the grip of the snake as close to the opening of the drain as possible. Continue to reposition the grip as the coil moves deeper into the drain. Eventually, the tip should reach the clog. When the clog can be felt or the coil will go in no further, you can begin to turn the handle, rotating the coil.
The rotation of the coil causes the auger tip to chew into the clog. It is usually necessary to move the auger tip back and forth during the part of the operation. In some cases, the clog might be broken up. You will be able to tell this because the tip will advance further into the drain. In other cases, the clog might be snagged on the tip. In this case, it will be possible to pull the clog back out of the drain. This is the preferred method. When the clog is not removed, it may reform deeper in the drain clogging it again.
If you do manage to pull out the clog be prepared for a mess. A toilet clog is not going to be the most pleasant thing. It is a good idea to prepare the bathroom, and yourself, for this mess. Make sure there is an ample supply of old towels and rags handy to protect the bathroom floor and to clean the snake coil as it is withdrawn from the toilet. Snaking a toilet is not a job that is done best in your best clothes either.