50mm - 55mm was considered the "standard" lens back in the film days because it gave more or less the same field of view as the human eye on a 35mm camera. Most of them were quite fast, and some were extremely fast.
With digital, it depends whether you're using a full frame body or a crop body. Full frame bodies have the same field of view as 35mm film cameras, with the same lenses, for all practical purposes. They're becoming more popular but are still expensive, and many (most?) of us use crop bodies. Most of them have a 1.5 or 1.6 crop factor, so a 50mm lens gives you a field of view equivalent to 75mm - 80mm, which is getting into short telephoto range.
The Nifty Fifty usually refers to the Canon f1.8 50mm. I've used it, but don't own it. The lens is cheap, and a lot of people think its fantastic value because it's fast and produces good results, but the build quality is very poor. The older Mk1 had a metal lens mount, but the current model is plastic.
Up to you really. Some people use it as a portrait lens, but I don't have any need for an 80mm lens. Too long, or too short, for my purposes. OTOH, it does get you a fast prime for an almost disposable price.
