Running shoes.
And I've gone through about six different brands of them. Asics and Saucony are my favorites. New Balance is pretty good. I had Brooks for a while in high school (they also make awesome running apparel). Nike and Adidas spikes back when I ran track, as well as some Nike Free Trainers when they were first released.
And I'd say that only runners attach the term "running" to this specific type of shoe. Otherwise, there's no real point. They're just a shoe, albeit (in my opinion) a more comfortable one. I usually buy a cheap pair as my everyday shoes, but I'll spend around $100 for when I need a good pair to run in.