Normally a toilet's bowl will drain completely when you press the handle on the tank, meaning something about the toilet is not operating correctly if your toilet's bowl does not empty out. Determining the cause of the lack of flushing power does not take long if you know what to look for.
Your toilet may be one of the first low-flow toilets made, which means that the toilet will not flush properly no matter what adjustments you make. To see if your toilet is one of the first generation low-flow models, remove the toilet's tank cover. On the inside of the tank you will see a manufacturing date stamped in the clay. If your toilet's date lands between 1994 and about mid-1997, you have a first generation low-flow toilet. There may possibly be a clog or other problem causing the toilet not to drain, but you should get a new toilet to avoid having as many problems in the future.
Before you start fixing things that are not broken, test out the toilet to see whether the problem has to do with the drain or some other part of the toilet. Pouring a large bucket of cold water into the toilet's bowl will help you determine if the toilet's drain is clogged or not. If there is a clog, the toilet still will not drain. If the problem is with another part of the toilet, it will flush just fine.
Unclogging a toilet requires you to use the right tools the right way. With weak clogs, the bucket test may completely or at least partially clear the drain. The bucket test will ensure there is plenty of water in the bowl, making for the right conditions to use a plunger. Do not use a regular plunger since it works better on sinks and bathtubs, but go with a funnel-cup plunger that has a rounded bottom. If the plunger does not do the trick, feed an auger or snake down the drain to break the clog up.
The water level inside the toilet's tank may be to blame for the bowl not draining completely. If you lift the cover off the tank and look at where the water sits, it should be about an inch from the top of the plastic overflow tube's opening. With too little water in the tank, the amount of water entering the bowl each time you flush is lessened. Adjust the water level in the tank by bending the arm that attaches to the float ball. If your tank does not have a float ball, twist the plastic screw on the top of the float assembly so it travels up the metal rod higher.
Clogged ports, which are located just below the rim in the toilet's bowl, will also cause the bowl not to drain completely. Cleaning out the buildup in the ports can be accomplished with an instrument such as a wood dowel or a wire coat hanger. You may also mix 1 part muriatic acid with 10 parts water and pour the mixture down the tank's overflow tube. The mixture should clear the ports out in about half an hour's time.