The door appears to be a veneered unit. The thin plywood veneer is breaking away. If it is limited to this one spot, it can be repaired rather simply. If there is more or the whole veneer is starting to delaminate, you should find a local trim carpenter to replace the door.
To repair, get your self a can of Minwax teo part wood filler. This works like Bondo but is formulated more for wood. It is essentially an epoxy. Follow the directions. You might want a pair of rubber gloves ( I use surgical type for stuff like this) or some acetone on hand for cleanup.
Cut away the loose material with a sharp knife, mix a wad of filler, spread it on , sand it out and paint it up.