The Curse Room is a new room type added with the Wrath of the Lamb DLC.
It usually contains one or occasionally two Red Chests. Sometimes, albeit rarely, it even contains a free Devil Room or Angel Room item instead.
It is connected to other rooms with a unique type of spiky door frame: walking in and out of the room will deal half a heart of damage to you. From The Womb onwards, it will deal a full heart of damage when walking in and out.
This room's map icon is a red half moon with a star.
If the Secret Room is adjacent to it, they will be connected with another spiky door.
If you enter this room while flying, you will only take damage one way.
This is also true when flying into it from a secret room.
The damage dealt is handled like normal damage done by enemies. (Soul hearts disappear before normal ones, etc.)
The items IV Bag, Razor Blade, and Blood Rights can be used as you're walking into and out of the room; this activates mercy invincibility, preventing the door itself from dealing damage to you. This essentially allows you to pay normal hearts instead of Soul Hearts to enter and exit the room if you have one of those items. Also if you have the My Little Unicorn item and activate when entering or exiting, the invincibility prevents you from taking damage.
Using the XX Judgement Tarot Card in a Curse Room and giving sufficient money to the Beggar will give you a Devil Room or Angel Room item.
The type of item you get (Devil or Angel) depends on which is present on the floor (i.e. receiving an Angel item from the room means an Angel Room is on the floor).
If you enter the curse room while on a spider web you won't take the initial damage for entering it.