Even when broken, removing door handles is simple. Whether you have a broken door knob or latch-type handle, the basic principles are the same. It used to be possible to fix door handles, but most locksets come as complete kits these days, and not all handles will fit all kits. It's best to just replace the whole thing once you have removed the broken handle.
Look for the screws that attach the handle to the door. Sometimes they are hidden beneath the decorative plate, called the rosette, that sits flush against the door. Check the shaft where the handle is attached to the rosette, too.
Remove a flush pin in a handle shaft by pushing on it with a straightened-out paper clip. Some locksets also have a wire securing the handle, but pushing on it with a small Phillips head screwdriver will remove it.
Try an Allen wrench if it looks like it will work. Every lockset is different, so you really have to look at how yours is put together, and you might also have to try several different tools before you find one that works.
Remove the handle or door knob once you have found and removed any hardware that was holding it in place.
Slide a thin putty knife between the decorative rosette and the door, and loosen it enough to slide it off over the knob or handle shaft if you can't see any visible screws attaching the rosette to the door.
Use the appropriate screwdriver to remove the lock plates on either side of the door.
Remove the small latch and catch plates on the inner edge of the door and the doorjamb. These also come in different sizes, so when you are buying a replacement kit, make sure it comes with the doorjamb plate, or purchase one that will fit your new lock.
Remove the hinge pins, and wiggle the door off of the hinges if the handle breaks off and locks you in.
Don't guess when buying a replacement lockset; take the old lockset with you to buy the new one.