Just about every toilet on the planet, except out-houses and airplanes, have toilets that are similar to the one above. On this toilet the trap goes toward the rear. Some models have the trap going up toward the front of the bowl before turning down to the drain. For the purposes of this instruction, it doesn't matter which type you have. The normal water level in the toilet is that of the blue line. The water must be at this level for some important reasons. If the water were too low, a lot of the flush would go to filling the bowl before the actual contents began to flow toward the drain. If the water were lower than this, sewer gases would be able to come up through the fixture and into the room. These sewer gases are not only stinky, but can also be mildly toxic, and under some conditions they can also be explosive! If the water level is too high it simply runs through the trap and to the drain.
If your toilet is plugged it will act in one of these two ways. The water will not go down at all and will overflow onto the floor when flushed. Or it will rise to the top of the bowl and swirl down slowly after a flush, often taking two or three flushes to get rid of solids and paper. The toilet almost always plugs in one of two places as noted by 'A'. Either some foreign object cannot make the bend at the top of the trap and becomes lodged there or it will make it around but won't go through the hole at the bottom of the toilet to the drain.
Most people think a plunger is a simple tool and they know how to use one. Most people don't! To properly plunge a toilet you must bring the water level to near the top of the bowl ('B') to get the hydraulic forces of the water working in your favor. This will cause there to be nothing but water between your plunger and the clog. No air that can compress causing there to be less pressure exerted on the plug. To get the water to this level if the bowl is plugged completely, remove the tank lid and lift the flapper by hand while watching the level in the bowl. When the water comes near the top, start plunging. Plunge back and forth as hard and as fast as you can for about six or eight repititions. If your feet get wet, you are plunging hard enough! The first few plunges will get rid of the remaining air and the next few should do the trick. It should all be over in five or six seconds. Lift the plunger and test flush making sure you have the tank lid off so you can manually push the flapper shut in case it is not unplugged and the water flows over the top. Repeat if necessary.
If the toilet remains blocked after a darn good plunging it needs to be augered. You can usually rent a toilet auger (toilet snake) very cheaply, or buy one for ten bucks. With the handle pulled all the way back you stick the end of the snake into the toilet trap and while turning the handle, feed the snake through the bowl. This can take a few minutes practice. Once you have fed the snake into the toilet as far as you can go pull it back and test flush again by manually manipulating the flapper. If this method fails the toilet may need to be removed.
To repair your toilet be sure to turn off all water and completely drain the tank and bowl before removing the toilet from the floor. Disconnect the water-line, unbolt toilet from floor and lift straight up and set on it's back carefully on some cardboard. You may be able to see from under the toilet what is in there and you may be able to grab it with a coat hanger or needle nosed pliers. I have snaked toilets from the bottom before and have been successfull. I have also cracked a few, so only do this as a last resort. If you will be rolling the toilet around while you are working on it, get help. Allowing the weight of the bowl to rest against the tank may cause it to break.
This diagram shows a toilet auger being twisted through the bowl. This picture shows the auger already threaded through the toilet. Start by holding the body of the snake in your lower hand with the handle pulled all the way up by the other hand. Feed the head of the snake into the base of the bowl. You must be twisting while pushing downwards on the handle to feed the snake into the bowl.
There are some things that can get into a toilet that swell when they are in the water and you won't be able to get them out. This is rare but bowl replacement is needed in one blockage out of fifty or so. If you have no success with the above, it may be worth contacting a professional plumber. Good Luck!