Oftentimes when a toilet stops flushing properly, it is due to flapper issues. The flapper is the plug that opens to let water out of the toilet tank and closes to allow it to refill. Though flapper problems can be annoying, they are usually easy to fix.
Adjust the Toilet Flapper
If you have a flapper that closes too quickly, this will prevent the toilet from completing a flush or leave your bowl with a low water level. To correct this issue, you can adjust the flapper in order to make it stay open longer. To adjust the flapper, take the cover off the back of the toilet and inspect the flapper chain. If there is a lot of slack in the chain, adjust it by unattaching the chain from the flapper (it should just pop off), pulling the chain taut, then loosening it by one link to create just a little slack and attaching that link to the flapper. Fixing the length of the flapper chain should also prevent you from having to hold down the handle on the toilet to complete an entire flush.
Replacing a Flapper
As flappers age, they wear down, allowing water to leak from the toilet tank into the toilet. This will cause a drop in the tank water level, which will make the toilet continuously refill, resulting in a continuously running sound. The only way to fix this issue is to replace the flapper. You will first need to purchase a new flapper. Make sure you know the make and model of your toilet before seeking a replacement flapper. Not every flapper will work with every toilet.
To replace the flapper, shut off the water supply. This will help prevent any major accidents. Once the water is turned off, you will also need to shut off the water that runs directly to the toilet. The valve to shut off the toilet's water supply is generally placed in the wall just behind the toilet. Gently turn the valve to shut off the water, then open the toilet tank and flush the toilet. Hold the flapper while the toilet flushes to allow all water to drain from the tank. Once the water has drained, remove the flapper from the tank. Nearly all flappers simply clip on to the hinges inside the toilet. Dry the inside of the tank with a towel before applying the new hardware. Press the new flapper onto the hinges. Pull the flapper chain up to the lever so that it is tight, go back one link and attach. Turn on the water supply. The tank will automatically fill with water. Flush the toilet with the lid off to make sure everything is working properly and the flapper is getting a good seal when it closes. If it isn't, you may need to replace the seal around the valve that the flapper rests on. You can buy valve seat repair kits that have peel and stick replacements for this seal.