What does it mean when the back of my hand starts to hurt?

Pain: That could mean a lot of things. It could be a nerve entrapment, muscle spasm or it may be related to the bony structures of your hand. Have it checked if you are having it for a long time already.

Many things: This reminds me of the old joke, "doctor, it hurts when I do this." "then stop doing this!" seriously, it could mean anything from a temporary strain/bruise to arthritis to infection to nerve damage. In order to sort out the cause, a doctor has to do a physical examination/tests to see what's up. See your doc if it lasts more than two weeks or is unusually severe.