Surgery is done most often when the skin is broken open and the bone is sticking out. This is called an open fracture. Closed injuries can be treated surgically or nonsurgically.
The type of fracture you’re describing is called a mallet fracture. A small piece of bone breaks away from the last joint in the finger. The extensor tendon attaches there and is also torn.
Without this tendon, the finger doesn’t extend or straighten all the way. It stays in a bent position even when you try to straighten it. This is called extensor lag.
Studies have shown much higher rates of problems when closed mallet fractures are treated surgically. They seem to do better with a splinting program. Patients report they are painfree and able to resume all their former activities.