The next figure to look at with a lens is the maximum aperture. Apart from a few oddities, lenses have a range of apertures they can do, from maximum to minimum. We don't care about the minimum. We do care, very much, about the maximum.
If you've skipped ahead to here, and haven't already read about Aperture, you might want to pop back before we go on. I'll wait here.
A 'fast' lens has a wide maximum aperture. There's no set definition of 'fast', and it varies with the focal length of the lens. With a 50mm lens, f/1.8 is pretty common, f/1.4 is quite fast, and f/1.2 is getting good. If you wanted a 200mm lens with an f/1.2 aperture, you're going to need a wheelbarrow full of money. And you'll need the wheelbarrow to carry the lens around after you've bought it, so don't . F/4 is decent for a 200mm lens, and f/2.8 is fast.
For me, it's a very important feature of a lens. The wider aperture it can use, the more I can limit the depth of field - the more blurry I can make the background. My 40mm f/1.4 Vöigtlander Nokton Classic can do it quite well, but so can my Leica Tele-Elmarit 90mm f/2.8. The longer focal length helps too, so a longer lens doesn't need as wide an aperture.
The wider the maximum aperture, the more expensive the lens will tend to be. Buying used vintage lenses can be a good way to get fast lenses at a decent price, but manual focus is harder when the depth of field is lower, and even very old lenses can have pretty high prices when they have a wide maximum aperture. Plenty of other people (like me!) are after those f/1.4 lenses too, which drives the prices up.
Depending on the type of photography you do, it might not be important to you. Street Photography tends to use relatively narrow apertures, to have plenty of depth of field. Landscape Photography isn't usually about shallow depth of field. Flowers, maybe. Portraits, almost always yes. Abstract and minimalist can vary - usually it doesn't matter too much, but sometimes it will.