A wide-angle lens captures a wide or broad area of view. A fisheye lens, one example of an ultra-wide angle lens, captures close to a 360-degree horizon.
A wide-angle lens has a focal length of less than 50mm. According to Heather Powell, writing for ePHOTOzine, "Landscape photography is where wide-angles tend to be most commonly used." A wide-angle lens may also be selected to shoot close-up pictures, often resulting in compositions with a distorted perspective.
A fisheye lens may be either circular or full-frame. A photo shot with a circular fisheye lens--having a focal length between 8mm and 10mm--results in a complete picture circle surrounded by black on all sides. A photo shot with a full-frame fisheye--having a focal length between 15mm and 16mm--has a more conventional, rectangular appearance.
Both general wide-angle and fisheye lenses have a short focal length. Shooting with a small f-stop (f22, for example) will result in incredible depth of field. This means that everything in the image, from foreground to background, will be in sharp focus.
The wider a lens' angle is, the greater the image distortion. Because a fisheye is an ultra-wide angle lens, the distortion is extreme.
Wide angle vs. fisheye
If you are looking for greater flexibility and more traditional images, then a general wide-angle is your best choice. According to photographer and author Scott Kelby, a fisheye is a "special-effects lens that you will want to use sparingly, because the fisheye look can get old fast if you use it too much."