if you have only damaged or broken off the head of a bolt and feel the threads should be intact i recommend the following methods.
go to your local hardware store and buy a lefthand drill bit that is about 2/3s the size of the bolt you want to remove and/or an automatic punch with a fine point tip. (i would get both to save a trip)
first find a high spot on the broken bolt using the automatic punch put a dimple on the back side of the high spot in order to try to rotate the broken bolt counter clockwise. once you have a decent dimple hold the automatic punch at about 35-45 degrees in order to use the force of the automatic punch to rotate the broken off bolt. it will take patients and technique but if this works you will not have to retap the hole (so long as the thread are not damaged).
if the aformentioned method does not work for you or sounds like too much work you can use the lefthand drill bit method. use a pin punch to make a dimple in the very center of your work surface (the top of the broken bolt) this will help you stay centered as much as possible so that you are less likely to damage the treads. in most cases a lefthand drill bit, a lil luck, low speed/high torque set drill and light touch on the trigger will do the job very nicely.
these are 2 generally low cost way to remove broken bolt. i have used both of these methods for a number of years and have had much success. i hope this helps you.