The toilet handle is a commonly broken part and the easiest part to replace on the toilet. It is also referred to as a tank lever as it is the lever in the toilet tank that causes the toilet to flush when it is pushed.
To replace the toilet handle, first you must remove the old one. The first thing that you will notice on a typical toilet is that the nut that holds the tank lever onto the tank will be reverse threaded. This is important, because many amateurs will try to take off the nut by applying counter-clockwise torque (lefty-loosey, righty-tighty) and they will just end up breaking it, as counter-clockwise torque will actually tighten the nut! The shank and nut on a tank handle lever mount are designed reverse threaded because over time of multiple use of the toilet handle, the nut would loosen up and come off if the threads were normal clockwise threaded.
How to replace a toilet handle
Once this bit of information is known, then the whole process is quite simple. Just twist the nut clockwise using a small adjustable wrench or pliers and it will loosen up and come off. Then unhook the flapper chain from the tank lever and remove the entire handle assembly. Now insert the new handle assembly through the toilet tank handle hole and screw the new nut on counterclockwise. Then clasp the hook off the flapper chain onto one of the holes in the new tank lever and you are done!
Usually this is also a good time to replace the flapper. Or at least follow the directions in that article on adjusting the flapper chain.