Photographers rely on standard workhorse lenses: the 50mm and 85mm portrait lenses are two of the most common "prime" focal lengths. Many photographers, including myself, prefer an everyday prime in the 35 or 40mm range, which is slightly wide but is arguably a closer approximation of human vision. Besides the primes, you have the zooms, useful for everything from sports to fashion to wildlife. This is an article about specialty lenses that you probably won't use every day, but are nevertheless useful to have in your bag once you've built your basic kit.
Fisheye lenses allow for photographing scenes of around 180 degrees. This is well beyond the normal range of unaided human vision, which spans about 120 degrees. The extreme angle of a fisheye necessarily introduces a level of distortion, which may be desirable (or at least tolerable) for certain subjects. Some fisheye lenses create a circular image in the middle of the standard rectangular photo frame that we're accustomed to seeing, whereas others crop that circular image to fit within the confines of a rectangle. The lenses themselves have a short focal length, typically around 8 to 10mm.* Fisheye lenses are typically less expensive than ultrawides. You can even buy inexpensive add-on filters for your camera or smartphone that will approximate the effect of a dedicated lens.
Ultrawide lenses offer something close to the wide angle effect of fisheye lenses, but tend to be a little longer. They don't curve lines the way that fisheyes do; the distortion they introduce can usually be corrected in image editing software. This difference makes them better suited to many professional applications. You get the advantage of an extremely wide perspective without the extreme distortion of a fisheye lens. Typical ultrawide lenses run from 11-16mm in focal length (regular wide angle lenses start around 18mm).
Contrary to what you might assume, the main purpose of these lenses is not to squeeze as much into your photo as possible. Of course they can be used that way if you need to do so, but without a clear subject the results will likely fall short of your intentions. From an artistic perspective, a better way to use these lenses is to get close to your primary subject, while showing background details that introduce depth and add interest. An example would be positioning yourself on the ground to photograph a cracked flower pot with an abandoned house looming in the background.
If your lens allows for very close focusing, you can also use it for comic effect by distorting a face - be it the face of a friend or your pet. Another fun use for these lenses caters to the universal selfie craze. Instead of including just one or two people in your selfie, you can easily crowd half a dozen people into your photo - all while holding your camera at arm's length! If you're interested in an inexpensive alternative to a dedicated fisheye lens, consider the Fisheye 2 camera made by Lomography. These are modern, plastic cameras that shoot round images on ordinary 35mm film. Point and shoot fisheye fun at its easiest: with a fixed focus of f/8.0, and you can get the film developed most anywhere.
In the event that you do want to use an ultrawide lens for a large group photo, keep your subjects well away from the outside edge. Be sure to position heavier folks nearest to the center. Also, unless you want to exaggerate distortion, be sure to keep the camera level. You'll see the most distortion the further away you get from the center of the frame.
Adobe Lightroom's lens correction feature can be a great help in these cases - just don't expect it to be able to straighten the wildly curved lines of fisheye lenses. An advantage to both fisheye and ultrawide lenses is that at any given aperture, your depth of field is still very deep, so beyond about 12" everything in the scene will appear to be equally sharp.
*The actual focal length of fisheye and ultrawide lenses varies depending on the kind of camera and sensor (or film) being used. For more detailed discussion on that subject, see this article. Also, the designation "ultrawide" varies between manufacturers; the general consensus is that it's very wide but doesn't suffer from the linear distortion you see in a full-fledged fisheye.