There are two types of lens distortions. One is called barrel distortion, and the other is called pincushion distortion.
Barrel distortion is typical with wide-angle lenses (<35mm). Notice the picture below.
This type of distortion presents itself as a bulge. Notice how the center of the frame is somewhat bulging out. This, usually unwanted, can be used for dramatic event in certain circumstances such in landscape photography. With barrel distortion, you can highlight the grandness of the sky.
The above image was taken with a full frame camera at 16mm.
The other type of distortion is called pincushion distortion. This is usually present with telephoto lenses(>75mm).
This image is a simulated example of pincushion distortion using the same image as before.
Distortion is most prominent when shooting straight lines such as with brick walls. Below are extreme examples of these types of distortions with straights lines on a test chart.