Description taken from Arcade version of this game

Mike Haggar has just been elected major of Metro City, a city that's been overrun with crime and gang violence. The gangs of the city, especially the powerful Mad Gear gang, have used bribery and intimidation to keep the former city officials in line, and Mike Haggar's next. The Mad Gear gang has kidnapped his daughter Jessica, holding her hostage in exchange for Haggar's cooperation. But Mike Haggar is a former street fighter, and he's determined to rescue his daughter and put an end to this threat to the city. With his friends Cody and Guy, he must fight his way through the city to the heart of the Mad Gear gang.

Final Fight is a 2D side-scrolling fighting game. Taking on the role of either Cody, Guy, or Mike Haggar, you must advance through six areas of the city clearing out the gang infestation. You will have to cope with a variety of thugs, ranging from firebomb-throwing maniacs to behemoth wrestlers. To defeat these thugs, you have an arsenal of punches, kicks, throws, and jump kicks at your disposal. When you're in a tight situation you can also execute a powerful attack that hits everyone around you, but at the cost of a portion of your health bar. The levels are filled with items that can help you such as food to replenish health, weapons to use against the enemies, or jewelry and cash for points. Beating thugs up or destroying items like barrels, signs, or phone booths can uncover these items. ~ Kyle Knight, All Game Guide