Slides look really cool when projected and especially when viewed on a light table. Slide film refers to any kind of reversal film which can be projected. There are ways to print these (actually right now it is "way" unless you try digital) called Ilfochrome which is rather expensive but the prints look great and are highly archival. Slide film is available in many formats and emulsions. The one you're probably referring to is 35mm slides. They can get mounted in the little 2inch universal mounts and projected in any 35mm slide projector. You can request unmounted for scanning or printing and they will be returned in sleeves or rolled.
Medium format slides are a whole different world. You can only get them processed unmounted these days. You have to cut them up and place them in mounts. You also need special expensive projectors. I have one and the results are very impressive, but obscure.
If you want to print in a darkroom, choose negative film for starters. The paper is very cheap and the process is shorter/less involved than slide film.
I guess the reason people shoot slides is the look. It's very impressive. Looking at a 4x5 slide on a light table is very very cool.
Kodak carousel 35mm slide projectors are literally being thrown out on eBay.