How do I stop water from coming out of the dispenser on the door of my refrigerator?
Sounds like the valve is broken.  You'll have to locate the shutoff valve, and turn it off there for now. 

Pull the fridge out, and look for a valve behind it.  If you can't find one, you'll have to look elsewhere (basement, crawlspace, etc.). It's common for fridges to be fed using saddle valves, so you might want to start looking for that first. 

You should be able to find replacement parts online, but you'll have to know the make and model of the fridge.