A plunger is used to provide suction, which will pop the clog clear and allow it to be flushed away. Place the cup of the plunger over the mouth of the drain and pump it up and down quickly several time. Flush the toilet; if the water drains away, the clog is gone.
An auger is also known as a plumbing snake, and it involves a long metallic cord with a corkscrew tip that is threaded down into the drain. The end of the snake cuts up the clog or drags it back up so that it can be disposed of. A closet auger is specifically made for use with a toilet.
You may also try to dissolve the clog. Pour a cup of that is equal parts hot water and equal parts dish detergent down the drain, wait 30 minutes and try flushing. Do not use drain cleaners, as they can corrode the pipes.