How can I remove a key that has broken off in a lock?
In the past I've used a thin barbecue skewer to extract broken keys, I also found a set of eyeglass screwdrivers handy as well. It all depends on how deep it broke off of course. 

EDIT:
How you use the skewer depends on how the key has broken. Keys are commonly made of cast lightweight metals which have a rough texture with lots of pits outcroppings along the break. What you are trying to do is use the point of the skewer to get into one of those and then use the bottom or top of the lock slit as a fulcrum to pry the key out.

Alternatively if you can get the point into the top of the lock above the key you may be able to get the point in front of the rearmost tooth and then use to the top of the lock slit for leverage. 

It may be useful to have more than one skewer, thin screwdriver, what have you available and use them in conjunction.