For a toilet that runs constantly or has a loose handle, look under the tank lid to ensure the lift wire is connected and unkinked, the flush valve is sealing all the way and the water level is not too high. If the toilet leaks water onto the floor, ensure the water connection and tank sealings are tight.
If the toilet runs constantly, remove the tank lid, and check that the flush valve seals completely. If the lift wire is kinked or knotted, or if the float ball rubs against the side of the tank, a small adjustment to the affected hardware fixes the problem. In some cases, bending the float arm slightly to lower the float ball removes undue tension that is preventing the flush valve from closing all the way. If the flush valve still does not seal, either the flush valve itself, the float ball or the ballcock may need replacing.
If the toilet does not flush when you push the handle, it's most likely that the lift chain is disconnected or that the mounting inside is loose.
For a toilet that leaks water onto the floor, check that all the connections are tight, including the water connection to the tank and the tank bolts. If the wax ring around the base of the toilet is worn, replace it with a new one. Alternatively, if water on the floor is a result of condensation during the summertime, insulate the inside of the tank to prevent this from happening.