The fisheye lens takes photos in a hemispherical fashion not simply spherical. That is to say, you aren’t just taking a circular picture, rather the view you are getting is closer to laying on your back and looking up into the sky and moving your eyes without moving your head to get all of the particulars from the landscape.
Some critics of the fisheye tend to refer to it as a distortion of an image. However, both rectilinear photos and fisheye lens photos distort images. The distortion is just determined by the lens by how the shot is distorted. In a fisheye lens, the distortion appears to be in the center (because of the apparent curvature), while in a rectilinear shot, the distortion is at the edges.
These fisheye lenses were the first to be developed. These lenses can actually make a photo appear as a globe. In fact, these lenses are usually used for cartography to give the photos a relative image to the curve of the earth.
These lenses allowed for the previously circular photographs to be expanded covering the entire frame of a roll of film. The biggest difference (and possibly disadvantage) is that putting a circular exposure onto a rectangular film is distortion on the edges.
The peephole in your door actually is outfitted with a fisheye lens. The lens allows you to see a huge scope of the area outside of your door. This makes the focal point right in front of your eye so you can better observe.