All demons have more or less the same powers; Psychic powers. Demons are able to control objects, making them move or break them. They can also easily fling a human across the room with nothing but a snap of their fingers. Some higher rank demons have these same powers, but in a far more powerful scale. White eyed demons have been known to demolish entire buildings within seconds.
However, while on holy ground within the cathedral, the demons have been weakened, and many are completely powerless.
All black-eyed demons have completely lost their power, and only the higher ranked ones have held on to some of it.
Lillith
Lillith is the most powerful demon of them all, and still have her psychic power, however weakened. She is strong enough still to throw a human across the room, but not with enough force to kill him, merely hurt him. She can also still control objects, but need to focus all her energy on one object, unlike earlier when she controlled as many objects as she wanted at once.
Alastair
Alastair also managed to hold on to a lot of power, and though he can not throw a human all the way across a room anymore, he can still control objects - one object at a time - within a certain reach of him (10 meters).
Azazel
Azazel holds the same powers as Alastair inside of the cathedral, and can move humans and objects within a 10 meter radius. He can control more specific things, like bloodcells, organs or other microscopic things. These takes a bit more power from him, and he needs to focus hard on the object. He likes to use this power to kill humans from the inside, just by manipulating their blood stream or similar things.
Calcifer
Calcifer lost all his powers inside the cathedral, however he still kept his ability to make deals with humans. For more information about demon deals see "Crossroads Demons" under Demon lore.