I've always wondered - are things like ice screws and pitons always single-use? Does climbing like this basically require you to discard hundreds of dollars of equipment with each face you climb?
Bolts (expansion or glue-ins) are heavily used for protection but the use is controversial in places. They are basically always left in permanently. Push here as been to limit use to areas where there's no other options and to place them carefully to minimize impact. There's a push to only use high-quality stainless (or Titanium for areas exposed to salt) for longest possible life.
The rest of the protections used, nuts, hexes, cams, ice screws, snow pickets, and some exotic devices are usually meant to be removed after use and minimize impact on rock. Although if someone needs to bail in a hurry, leaving expensive gear behind is normal.
Nuts and hexes are basically blocks of aluminium on a wire or webbing sling. Nuts being smaller solid chunks and hexes being larger and hollow. They have tapered shapes (sometime complex, curved, or even designed for a camming action) and you try to wedge them in place. They can be very secure in the right placement. After a good fall on one, it sometimes happen that it cannot be dislodged and it stays fixed for a few months or years.
Cams are mechanical devices designed to grip both sides of a crack and expand outward when loaded. They have two spiral shaped lobes to accomplish this. Very strong when well placed and they have a range of sizes to fit different cracks from tiny body weight only cams around for cracks of 1/4" or so to monsters that fit in cracks of 6" or more.
Ice screws are hollow threaded tubes, the article here does a good job showing how they function.
Snow pickets are T-shaped beams, about 2 feet in length or more that can be driven into hard snow or buried in a trench. Not that strong but workable.
There's a lot of special purpose "protection" that is usually barely able to hold body weight that gets used for aid climbing (when you use the gear to make progress) in situations where nothing else will work. Often to link sections (called pitches) of unaided climbing (called "free" climbing). Free climbing still uses protection in most case, unroped "free-solo" being the exception.
You may come across gear given another name - such as Hex, Micronut, Stopper, Chock, etc - but they're all variations of the two: either a lump of metal that wedges into a crack in the rockface, or a spring-loaded movable metal device, that wedges into a crack in the rockface.
Nuts and cams (as well as ice-screws) can be easily removed by the last person on the route (who is being belayed by someone higher up, so doesn't need the protective gear any more). In most cases, gear isn't left behind, unless something's gone wrong.
[1] https://en.wikipedia.org/wiki/Nut_(climbing)
[2] https://en.wikipedia.org/wiki/Spring-loaded_camming_device
Also, pitons are rarely used these days except for in some types of big wall climbing. Typically rock climbing protection uses cams and nuts, which don't mar the rocks the way pitons do.
In Europe, pitons may still be used in some limestone areas that don't take modern gear very well.
No. Ice screws are definitely something the follower cleans (just unscrew them) and pitons can generally be removed, either by hammering them back and forth, prying them out like a nail, or using a funkness device (a wire with a loop on each end) to yank them out. "Big Walls: Elite Technique" by Jared Ogden has some nice diagrams, but they didn't turn up online when I searched for them. Also, pitons generally aren't used any more (definitely not in sport or trad climbing), but I wouldn't say they are niche or rare. Many big walls in Yosemite still haven't been done "clean", they still require the use of pitons.
This is coming from someone who's only climbed clean aid and whose climbing partner really doesn't want to climb anything that requires a hammer, but I've still picked up a hammer and some old pitons to practice with.