There's no need to explain too much. Mortal Kombat changed everything. It brought a new button scheme (two buttons for punch, two for kick, and one for block), "photorealistic" graphics, a lot of blood and violence, and, of course, the great and infamous fatalities. Such was the success of Midway's game that, just like what happened with Street Fighter 2, the trend took over and many other games began to imitate it, many including some type of "finishing move".
Here are some of those clones, the craziest, strangest, and bloodiest.
Barbarian (1987)
Those of us who are older than we dare to admit know that Mortal Kombat was not the first video game to include bloody finishing blows to finish off the opponent. Oh no! The first fighting video game to include a fatality was Barbarian (for C64 and other systems). The good thing about this "fatality" is that it could be executed at any time, and not only when knocking out the enemy. You just had to land a spinning blow at the right moment and you cut off his head. Then an ugly lizard came and kicked it off the stage.
Eternal Champions: Challenge from the Dark Side (1996)
Eternal Champions, the Sega CD version in this case, had two types of finishing moves: the stage fatalities and the vendettas. That is, there is one fatality per stage, which can be activated by any character, in addition to the character's unique fatalities, called vendettas. The game is more or less, but with very nice graphics.
Weaponlord (1995)
The fighting game for Sega Genesis with character designs by Simon Bisley was not as good as it should have been, but... it had Simon Bisley's designs! Weaponlord's art is exceptional, though the same cannot be said about the fatalities, which are as mediocre as its gameplay.
Tattoo Assassins (1994)
One of the strangest Mortal Kombat clones. Developed by Data East, created by the writer of Back to the Future, and with different types of "finishing moves" to choose from. Besides the fatalities you can see below, the game had Morphalities and Nudailities (in which the losing character is stripped naked).
Survival Arts (1993)
Another one of the cheap Mortal Kombat clones, but with controls very similar to those of Street Fighter 2, just to confuse. It wasn't better than either of the two, so it faded into obscurity without much glory. Today we remember it here, for its fatalities.
Time Killers (1993)
Time Killers possibly has the best pixel art graphics among all the games of that era. Finally a bit of color in a fighting game! Besides fatalities, in Time Killers you could lose body parts and keep playing, as long as you had health points. A gem of a game.
Thrill Kill (1998)
The forbidden PlayStation game. A game so bloody that it was never officially released, although we all played it thanks to the leaked beta. Thrill Kill was insane and, although the PlayStation's graphics didn't help it much, also very fun. These are the fatalities of all its characters.
https://www.youtube.com/embed/uELqQjkQNawWay of the Warrior (1994)
What a terrible console the 3DO was, Panasonic's abomination that tried to compete against the PlayStation. It's possible that at a hardware level it didn't suck as much, but the games were definitely among the worst. Look, for example, at this disgusting Mortal Kombat clone, with graphics taken from some B-movie and trying to pass as more than a miserable copy.
Primal Rage (1994)
The dinosaur game developed by Atari had everything in its favor to succeed, except for a terrible gameplay that made it too mediocre next to the great titles it competed with. It's a well-known game, but not much played, so it's possible you missed the fact that it also had "finishing moves". They weren't that great either, as you can see in the video.
Terrordrome The Game - Rise of the Boogeymen (2012)
In Terrordrome, the fighting game based on classic big-screen slashers, the fatalities are called Trophy. But they are the same thing, a violent finishing move to finish off the loser in the bloodiest way possible. Unlike classic fatalities, in these Trophy the winning character lifts the adversary's head as a trophy, reveling in evil.