Will a car start with a broken starter?

Depending on how broken it is you can sometimes hit the starter with a wrench or something hard and it may start.

Pretty much by definition, if the starter is broken the car won't start by using the starter, although the above can sometimes work as a quick fix. However, the car will start without using a starter, especially if it is a manual transmission,

Turn the key on, and push the car in neutral (even not very fast). Put it in gear and release the clutch while it is moving, which turns the motor and starts the car. The same can be done with most automatics, but you must push them much faster.