I agree with Luke, but there is an additional use as well. Some teachers when dealing with unruly students tell them to go and stand "in the corner" (they usually don't say - in the corner of the room -- as that is obvious).

I would most often say "on the corner." Example: "The man is standing on the corner by the convenience store." However, if you are giving directions, you can say both "It is at the corner of 5th Street and Madison" or "It is on the corner of 5th Street and Madison." Either one is fine.

The corner of an intersection or the corner of a room? Does the corner represent an area, or is it a point in a longer journey? Which preposition you need depends on that.