Generally, barrel distortion occurs when a wide angle lens is used to photograph a subject with strong vertical or horizontal lines. Basically a convex mirror effect is applied because the lense is actually bending the light, forcing a larger image area to be recorded on a smaller capture service. Or in super confusing mathematical formula format: R=(a* r ^ 3 + b * r ^2 + c ^r + d)*r. Simply, objects closer to the camera appear larger and straight lines are curved and objects in background appear smaller and further away.
On a 35mm format camera a focal length of 50mm is almost identical to that of the human eye. So, coincidentally what you see is what you get. Furthermore, I strongly believe every person interested in photography should learn on a 50mm lens first before moving on to different lens types. Any lens with a shorter focal length will cause barrel distortion.
Shooting a panoramic image allows you to use a a lens with a longer focal length yet still capture a wide field of view. I will often shoot two or three image pans so I can create the exact image I want without adding distortion.
When shooting with a wide angle lens always avoid putting subject matter in the fringes of the image. Barrel distortion increases exponentially as you move away from the centre of the lens. Photographing a group shot with a wide angle well often turn one or two members into a conehead.
Just don't. A Fisheye looks good in maybe 1% of the instances you will have the opportunity to use one, and even that is a stretch. They are expensive, make subjects look terrible and provide poor optic quality, so pretty much useless.
Preventing and properly managing distortion is a key step in the advancing the quality of your imagery. It is one of the many attributes that defines the quality of a professional photograph.