In reply to Juggler13: Ive only been climbing for a couple of months and that always used to happen to me but I managed to do it on every single finger joint. I found that when i couldnt go climbing for about 2 weeks due to being on holiday, my hands had time to heal properly and now they seem to be tougher. But i find that bouldering on overhanging problems still makes them abit tender so i try to stop before i get to the skin being pulled off stage.
I also found that when the skin was like that putting tape on made it hurt alot less for climbing (but the guy in the shop told me that this isnt what tape should be used for).
So I would suggest that if you really want to go climb then put some tape on, but if you can wait then rest your fingers and take it easy on them until they toughen up.