Although it is frustrating when you are unable to unclog your toilet with a plunger, there are several other methods that you can try without calling a professional. Different tools may be required to clear the blockage, depending on the nature of the blockage. If the clog is caused by a solid object, such as a toy or toothbrush, the toilet will flush perfectly with only water, but clog with toilet paper. The toilet paper catches on the item, causing the clog.
Insert the auger end of a hand-cranked auger (pipe snake) into the toilet bowl, pushing the auger so that it follows the bends of the waterway inside the toilet. Turn the auger's handle to aid the insertion. If you feel an obstruction, keep turning the handle until the auger breaks up the blockage. The auger will be able to break up a toilet paper clog, but if the obstacle is a solid item, such as a toothbrush or toy, the auger will not dislodge the item.
An expandable bladder will seal off the toilet's opening and force a high-pressure stream of water into the toilet's inner waterway. Attach the expandable bladder to a garden hose (if you can get it into the house), or other hose attached to the sink. Insert the bladder halfway into the opening and turn on the water. The bladder will expand as water is forced into the waterway. Some water may back up around the bladder, so be wary of overflows.
At this point, the blockage is definitely a solid item. Using a bolt grabber (available at automotive stores) or a bent coat hanger, try to grab onto the object and pull it out of the toilet.
You can then try to dislodge the item from the other end of the toilet. Drain the bowl and toilet tank, unhook the water line, pull out the toilet and carefully flip it on its side. If you can see the obstruction, insert a fish-hook remover to grab the item. If you cannot see the object, try to reach the item with a bolt grabber, which has a longer reach.
Take the toilet outside and lay it on its side. Insert the hand-cranked auger in the bottom of the toilet up towards the bowl. If possible, while you crank the auger, insert a water hose into the bowl and turn on the water. If this step fails, you will have to replace the toilet.