I have taken the back plate off of the lock and nothing appears to be worn or broken. However, the door was shut when the latch got stuck, so I cannot pull the whole latch out. The lock is made by Schlage. Please help
The lock might be broken, but it's also highly possible that it's just jammed in the strike by misalignment.
Pull the hinge pins (hopefully they aren't security pins) and pivot the door the opposite way to remove it. (If you can't remove the pins, try and pry the door up or down with a thin prybar to see if the latch will move.)
From there, reassemble the lockset and see if it works. If the lockset works, then it's simple alignment. Remove the strike plate, reassemble the door and figure out where the strike plate is really supposed to be.
Top tip: fill old screw holes with toothpicks covered in glue. When you're ready to put a new screw in, predrill with a small (~1/16") drill bit to prevent wandering.