Sprites are 2D graphics used in video games! The word typically refers to pixel art graphics, but technically refers to anything 2D.
A whole bunch of things!
UI and design! Pixels of the right size are nice and readable for design interfaces, and decorating your webpage or desktop with sprite characters has been a long-time online trend, from KiSS dolls to the avatars of Gaia Online...!
Comics! Sprite comics were always pretty niche and have largely fallen out of fashion, and largely relegated to the webcomic sphere, but it can't be said it wasn't a big draw at one point.
Plain ol' art study! Working in pixels is as much a medium as any other artistic outlet, and to just plain admire what folks have crafted with them, be it in screenshots, animated gifs or sprite sheets, is pretty valid in my book.
I've been ripping and playing with sprites for over half my life, so I may or may not be a mite biased in my evaluation. The point stands that while "sprite culture" of olde was kind of a nerdy teen thing to partake in, pixel art's gotten more in vogue the past decade-plus, and a respectable skill to have under your belt for an artist, a crash course in animation, readability, and making the most of limitations.
There's a lot of ways, and it depends how exhaustive you want to be! If all you want is a few poses for reference, taking screenshots from an emulator will usually suffice. Emulators come with features that can aid this process, like speed adjustments to advance frame-by-frame, or the ability to disable background layers.
If you want to get even more hands-on, they typically include RAM tools to tinker with the game's memory on the fly, allowing you to move objects on-screen, warp between levels, or even load new sprites and animations if it's cooperative enough.
But if you want to be more comprehensive, then you might want to look into the raw graphic data. Tile viewers are programs that can 'read' the tiles that comprise each sprite, and all that needs done is to piece them all together. Though you're at the mercy of whether this data is compressed or not (and whether there's tools to crack that compression), these tools can offer insight into how the game works and stores its data, spotlighting content that doesn't appear in-game,
Neither method is without its drawbacks, and it all varies on a game-by-game basis, but if you're passionate and patient enough they can both be gateways into learning new toolsets. Take a goosey into in-game ripping or tile ripping and see what happens! It's what the getting started section is for!
The unspoken emphasis in this section is that it's about how I rip sprites, and what feels natural to me might be complex, unintuitive, or outright backwards to other folks. Everyone has their own method, and lord knows I am extremely behind the times on technological advancements.
My sprites page occasionally has notes on specific games that needed special methods to be ripped. Do peruse the Sprite Database, VG Resource Wiki, or relevant forums for more information on tools, methods, and general know-how.