They asked questions on why you thought you would be a good fit for the camp. This includes how you would interact with the children and what you'd bring to the table. I was interviewed by the director, and the two coast directors - the camp is divided up into the East Coast and West Coast and it is by age groups.
It is fairly competitive for a summer job, but they have to be because it is one of the larger and well known day camps in the area. Since you'll be dealing with children all day long, they have to be sure of their picks.
On a side note: I had worked in their CIT (Counselor in Training) program, which you paid to do, because you're technically still a camper at that point. I interviewed for a Junior Counselor position, and from there I was asked back each year as a senior counselor.