Why screensavers existed at all
Cathode-ray tube monitors had a real, physical problem: leave an unchanging image on screen long enough and the phosphors that had been lit longest would burn in, leaving a permanent ghost of your menu bar behind. A screensaver kept the pixels moving so nothing stuck.
That was the entire technical justification, and it mattered from the early days of the color CRT until LCDs took over in the mid-2000s. Somewhere in the middle, the utility turned into a culture.
The Macintosh lineup
After Dark (Berkeley Systems, 1989) is the one everybody names. It wasn't a single screensaver but a collection of modules — Flying Toasters, Fish, Warp, Bad Dog, Rain Storm — with more available in later volumes and from third parties. It was sold in a box, with real design behind it, and it set the expectation that your screensaver said something about you. The full story of the toasters is here.
Before and alongside it there was a healthy shareware scene of single-purpose savers passed around on floppies — the kind of software that arrived with a text file asking for twenty dollars and an honest conscience.
The Windows lineup
Flying Windows was the default: the Windows logo streaming toward you through black space. Most people never changed it, which is exactly why it's the one that shows up in every nostalgia thread.
Mystify Your Mind drew colored polygons that bounced around the edges of the screen, trailing after themselves. Beziers did something similar with curves. Both were built in, both were free, and both were more interesting than they needed to be.
3D Pipes arrived with the OpenGL screensavers in the Windows 95 era and became a minor legend — a plumbing system assembling itself forever, in colors nobody chose, consuming more graphics horsepower than most games of the period.
Marquee scrolled a line of text across the screen. It shipped blank, which made it the one screensaver everyone customized, usually with something unwise, on a machine at work.
After Dark for Windows followed the Mac version in 1991, bringing the toasters to the other side of the platform war.
The one with a plot
Johnny Castaway (Sierra On-Line and Dynamix, 1992) billed itself as the world's first story-telling screensaver, and the claim holds up. Instead of a looping animation, it ran an actual narrative about a man stranded on a tiny island, unfolding across roughly 120 days of idle time. He fished. He jogged. He built things that fell over. Mermaids and pirates showed up. Holidays triggered scenes off the system clock, and the story eventually reached a real ending.
Almost nobody saw the whole arc, because seeing it required leaving one computer alone for four months in 1992. It remains the most quietly ambitious thing anyone ever did with a screensaver.
The Windows XP exception
By 2001 the interesting visual real estate had moved. Windows XP's default wallpaper — Bliss, a green hill under a blue sky, photographed by Charles O'Rear in January 1996 — did the job a screensaver used to do: it was the image your machine showed the world when you weren't using it. The difference is that nobody chose it. It came with the box, on hundreds of millions of machines. There's merch for that hill.
What killed the category
Three things at once. LCD panels don't burn in the way CRTs did, so the technical reason evaporated. Power management got good enough that the operating system simply switched the monitor off instead. And then phones arrived, and people stopped leaving a computer sitting idle in a lit room at all.
Modern systems still ship screensavers. The macOS aerial photography is genuinely beautiful and essentially nobody has ever chosen it on purpose.
What the nostalgia is actually for
Not the software. Nobody wants Flying Toasters back as a functioning utility. What people miss is the era it belonged to — when a computer was a thing you personalized, and the screensaver was one of the few design decisions a normal user made that other people in the room could see.
The design space of "what my computer looks like when I'm not using it" is basically closed now. It was open for about fifteen years, and this is what was in it.
Wear that era: the Screensavers collection, After Dark, or the Flying Toaster hat specifically.