Although I haven't really talked much about beat em ups except as a kind of aside, since so many Final Fight characters made their way over the years into Street Fighter, that's probably not really fair. Beat em ups came out about the same time as fighting games; maybe just a bit earlier, and their origins and development are inextricably entangled with that of fighting games too. If fighting games have a number of "fallow" years in the early 00s, the beat em up genre experienced a much longer fallow years blight; from the mid-90s until just a few years ago, honestly. Right now there's a fairly good scene for beat em ups, although to be fair, many of the titles are kind of indie and you may not have heard of them, but the revival of Streets of Rage in its belated fourth entry, brought the revival into the mainstream; it's available on pretty much every major platform right now, and has sold over two and a half million copies (that number is as of six months ago, and before the release of its DLC. It's probably significantly higher now.)
Because beat em ups were popular prior to the arrival of fighting games as we really know them, I was a huge fan of them prior to playing fighting games. While the narrative is that beat em ups suffered in popularity following the release of Street Fighter II, the reality is that probably the most popular and often considered the best one was released after Street Fighter II, and took deliberate inspiration from it. Beat em ups and fighting games followed a lot of parallel development in terms of how they should play, which admittedly could only go so far because of the fact that one is focused on 1x1 martial arts fist fighting in a duel-like environment, while the other is focused on beating up wave after wave of much weaker enemies (except the bosses) on a belt-scrolling stage. I think people stopped playing them because they weren't available anymore in arcades rather than because people didn't want them. I mean, when semi tongue in cheek fan project Beats of Rage gets a million downloads with only word of mouth, that strongly suggests that there was a market for good games of this type all along.
I already mentioned how the environment and enemies is a major difference between fighting games. The belt-scrolling stages where you fight wave after wave of arguably weaker opponents to end the stage against a boss that's comparable to you, if not stronger is the major distinction, but other things are significant too. The potential for cooperative play between more than one player is another key element, as is the use of disposable weapons like the knives, metal pipes and other things that you pick up and use for a little while before they disappear. The most iconic titles are fist-fighting martial arts games, like the most iconic fighting games, but the moves are much simpler, usually focusing on only two or three buttons. Over time, this has changed somewhat in cross-pollination with fighting games; more complex combos and special--even super moves--have entered the lexicon of beat em ups. But the originals didn't really have too many of them. Despite the fact that there were only a few buttons, there were combinations of things that you could do with them that led to more moves being available than merely two or three, of course--but the classic beat em up is a simpler game to play than the classic fighting game, at least in terms of player character moves.
From a story and setting perspective, there have been lots of different types, but the classic involves fighting street gangs in a setting of urban decay and chaos, often to go rescue one of the characters girlfriends. The first game that is usually called a beat em up in the traditional sense, although it still lacked the cooperative play feature, is "Hot-Blooded Tough Guy Kunio" which was translated into the West as Renegade. It wasn't just translated, though--it was graphically remade, and the whole rescue the girlfriend story was added. Rather than being a Japanese high school delinquent protecting his nerdy friend from being picked on by gangs from school, it was loosely based on the cult film The Warriors, released in 1979, which I don't think much of anyone saw at the time, but which has been referenced by creative types in multiple genres over and over and over again. (The music video for Shake It by Metrostation, for instance, is also based on it. K-Traxx and other EDM artists have sampled vocals from it in many places, and Renegade, among other video games, lifted the look and concepts straight from that movie too.) Renegade is an insanely influential game, but the beat em up as we all know it really was created the next year (1987) with the same creators building on both the Japanese and Western versions of the game to create Double Dragon. Curiously, the same year as the initial release of Street Fighter, which although a different kind of game, was obviously influenced by the success of Renegade in its title and concept.
Double Dragon was originally going to be the actual sequel to Kunio/Renegade, but the decision was made to give it a different cast, setting and make it its own thing, which was probably a good move. This is where the genre really finally came into its own; Double Dragon introduced the cooperative play, the continuous side-scrolling, the disposable weapons, etc. It also utilized the Western Renegade type setting and plot, including some cut scenes, and it went even further. Rather than simple urban decay as in The Warriors, Double Dragon takes place in an almost post-apocalyptic urban setting, heavily influenced by the Mad Max movies and Fist of the North Star. Much of these tropes would be copied by most of the other subsequent beat em ups. Heck, even such esoteric details as an enemy female character who dresses like a stripper and uses a whip were carried over at least into both Final Fight and Streets of Rage from Double Dragon.
Double Dragon was wildly successful; in America it was the highest selling arcade game for two years in a row, 1988 and 1989 after its initial release. By 1989, two rival franchises became probably the most successful in the genre, Sega's Golden Axe, which took the concept of the Double Dragon beat em up, wedded it to the developers love of the Arnie Conan movies, and the concept of special moves that Capcom had introduced in Street Fighter I. This game was hugely successful and spawned several sequels and spin-offs. I personally probably played more of it than I did Double Dragon, although I played a fair bit of Double Dragon in arcades in 87-88 or so. This type of game was even better served, I think, when Capcom got a hold of the Dungeons & Dragons license and created Tower of Doom and Shadow Over Mystara in the mid-90s, using big, Final Fight-like sprites.
Capcom, in fact, had a large run of beat em ups. Not only the D&D games, which are fantastic, by the way, but the weird post apocalyptic Cadillacs and Dinosaurs, Alien vs Predator, and futuristic superhero Captain Commando. The D&D games themselves probably came about because of Capcom's own King of Dragons and Knights of the Round. But probably the most famous, and later a contender for the top series of the genre was Final Fight. Originally conceived as Street Fighter II, albeit a totally different kind of game more closely based on Renegade and Double Dragon, Final Fight brought much better graphics, smoother gameplay and much more character than any game in the genre had heretofore delivered. The arcade version is a great game, but unfortunately, it got a SNES exclusive conversion that was hacked in many ways, and pretty disappointing. It's readily available now in modern systems; you can buy the arcade version on the Capcom Classics Collection (PS2 era) or the Capcom Beat em Up Bundle on Steam, PS4 Nintendo Switch, etc. (It's interesting, isn't it, to see this classic era of video gaming make a resurgence as gamers are discovering that the direction the market went was not actually in the direction of games that were necessarily more fun. Stuff that's now twenty or even thirty years old is being rediscovered and it turns out that it's considered good design even still.)
Final Fight was enormously successful; it was the most successful video game in arcades in 1990 (it's release date is late 1989--around Thanksgiving) and the second most successful arcade game in 1991, behind Street Fighter II. It also was one of Capcom's best-selling games on the SNES, in spite of the poor conversion. It epitomized the genre in almost every respect; the urban decay, the go beat up gangsters to rescue a girlfriend, the roster of characters which includes (admittedly somewhat generic) Cody, the boyfriend, but also the girl's father, Mike Haggar, who was also mayor, and Cody's best friend Guy, a Japanese ninja who was hanging around town for whatever reason.
Final Fight characters had so much character, in fact, that they've had a somewhat startling level of longevity. Even relatively "faceless" mooks have gone on to become Street Fighter characters (Hugo and Poison, in particular) while many of the bosses have as well (Rolento, Abigail, Sodom) and both Cody and Guy have appeared in multiple Street Fighter roles. Even some characters from the sequels (Maki, Lucia) eventually made their way into Street Fighter. Also extremely curiously, Mike Haggar has never made an appearance as a playable character, even though he did so in Marvel vs Capcom 3 Ultimate. Why they didn't port him to Street Fighter IV's later updates is still a frustrating mystery to me.
Although they are significant in Street Fighter as playable characters, they also don't really contribute in any way to Street Fighter's story, however. In fact, in the General Story mode of Street Fighter IV, I'm pretty sure that not a single Final Fight character makes so much as an appearance, even. They're part of the same continuity, but somehow part of a completely parallel existence.
Final Fight was also caught up in the console wars between Nintendo and Sega of the 16-bit generation. As I said above, Final Fight got a SNES exclusive port, although it had to make significant sacrifices from the arcade version to be ported to the SNES. While it was successful as a seller on the platform, I suspect most gamers were disappointed in it. Sega, seeing an opportunity here, took their Golden Axe engine and made an urban Final Fight copycat game which was exclusive to their console, the Sega Genesis, called Streets of Rage. While graphically it doesn't really compare to Final Fight, in most other respects, it is a better game than the SNES port, if not the arcade version, of Final Fight.
You'll notice that many fighting games from the early years followed exactly the same tropes as Final Fight and other beat em ups. Art of Fighting has the exact same plot. Fatal Fury has Geese knocked off of a skyscraper, just like Belger, the boss from Final Fight. Street Fighter Alpha 2 foregoes the fighting championship, and has characters moving around not unlike a beat em up in nature.
Final Fight 2 and 3 were SNES exclusives, not arcade games, and "fixed" a number of issues that the SNES Final Fight game had had, but Sega was not resting on their laurels either, and Streets of Rage 2 and 3 are considered by many fans to be the absolute best games in the beat em up genre; Streets of Rage 2 in particular. (3 is often seen as good, but offering little than 2 didn't already offer.) In fact, the game is often considered one of the best video games of all time period, which is really saying something. It continues more of the cross pollination that was happening with fighting games by introducing more moves and more variety for playable characters, something directly inspired by Street Fighter II and its success. One designer even said that having the same kind of combo rhythm as Street Fighter II was a key design attribute that they wanted to corner. Streets of Rage 2 also had greatly improved graphics over its predecessor, comparable to Street Fighter II's in most respects. In addition to this, Streets of Rage had what was considered at the time--and maybe even still--one of the greatest original soundtracks ever bundled with a video game. (Personally I still greatly prefer Star Wars the Old Republic's.)
It's curious to me that Capcom's big rival in the beat em up was Sega, while their big rival in the fighting game was SNK. SNK also had a Final Fight-esque beat em up called Burning Fight, which is capable enough, but which lacks a lot of the character that made Final Fight or Streets of Rage so memorable. Still, it was relatively successful in Japanese arcades, at least, when launched in 1991. I'm a little surprised that a King of Fighters '94 team wasn't made of the three selectable characters, to stand next to the Fatal Fury and Art of Fighting characters. Not that the USA Sports Team doesn't have its fans, but I think this would have been a better use of that team's slot.
Burning Fight has two American policemen from NYC come to Japan in pursuit of a Yakuza criminal gang that was causing trouble at home. They meet up with a Japanese detective and the three of them are the selectable characters. Among other things, if they had used these characters, King of Fighters would have it's own character named Ryu, although he would have looked and probably moved much more like Guy.