Fighting games are games in which two or more players fight with each other or against computer-controlled characters, often using martial arts. There is a variety of moves available, like punching, kicking, grabbing, throwing, and blocking, that should be used to defeat the opponent.
Most of these games are based on duels on fixed screens, such as Kung-Fu: The Way of the Exploding Fist. Progressive games like Double Dragon or Streets of Rage, where you need to fight multiple enemies, are usually defined as beat 'em up or brawlers.
Most of the first fighting games in the 1970s and 80s, like International Karate +, were about sports: boxing and martial arts. Then, some games like Barbarian: The Ultimate Warrior added more imaginative features that allowed the genre to grow more.
The end of the 80s and early 90s is when Street Fighter and Mortal Kombat were released. Both of these are considered staples of the fighting game genre. Street Fighter II, in particular, was a revolution with its accurate controls, simple gameplay, and multiplayer mode. The ability to play against other players really kickstarted the competitive side of fighting games.
While the concept of a fighting game itself is simple, it allows for a ton of freedom. Different environments, perspectives, combos, and even weapons. All that and more made the fighting games we know and love so popular. If you like this genre, don't forget to check brawler games too!