The King of Street Fighters

In 1991 Capcom released a sequel to their game Street Fighter called Street Fighter II: World Warrior. This game would turn out to be one of the most influential games in the "fighting" genre of video games.

Here comes the King

Capcom's Street fighter II is the most influential and copied game in the fighting game genre. It can even be said that the reinvented the genre. The original Street Fighter never made the same impact as its sequel. The one-on-one fighting game with its unique characters, different fighting styles and special moves made a big impact in the arcades.
From its release to 1993 sales revenues for Street Fighter II were more than 1.5 billion dollars.
But more importantly was the impact the game had on video gaming. Street Fighter is one of the most copied games ever, following the success of the Street Fighter franchise came an immeasurable amount similar fighting games.

The Competition

There almost isn't a video company that hasn't released a fighting game similar to Street Fighter. A lot of companies tried variations on the theme by adding weapons, animals, cartoony characters or some other unique selling point. But in the end they were all clones of that one fighting game.
Not that it mattered because the formula where you take your character and fight against a human or computer controlled opponent in a one-on-one battle proved timeless.
At that time there have been many attempts to surpass the original but all attempts failed. Midway probably had the best contender with its gory fighting game Mortal Kombat.

Mortal Kombat was a huge success with its more mature take on fighting. The fatalities and over the top violence were a big hit in the arcades. Mortal Kombat was very different from the cartoony characters of Street Fighter and presented the gamers with (for the time) life like characters.

Another company that got it right was Nintendo. Its rendition of a fighting game was called Killer Instinct. It was unusual for Nintendo to release this kind of game. They were best known for their Mario and Zelda games which were more directed at an audience of all ages.

Killer Instinct was a cross between Street Fighter and Mortal Kombat. It was not as violent as Mortal Kombat but looked a lot more realistic than Street Fighter.

SNK, the people behind the NeoGeo (Here more on SNK), specialized themselves in making fighting games.
They have produced some of the best 2-D fighting games besides Capcom.
Some honorable mentions are:

Samurai Showdown II
This game was set in medieval Japan and added weapons to the combat. It used zooming techniques and even had its own version of fatalities.

Last Blade 2
Also set in Japan, this is probably one of the most difficult and sophisticated fighting games. It never became as popular as the other fighting games from SNK like King of Fighters or Fatal Fury.

King of Fighters
Probably the most well known series from SNK. In this game SNK characters (also from other SNK fighting games) fought in a tournament to decide once and for all who was the strongest. There were characters from the Fatal Fury series, Samurai Showdown, Art of Fighting but also new characters. The character roster for King of Fighters has always been impressive.

SNK has a lot more great fighting games like:

Kizuna encounter
Fatal Fury
Garou: Mark of the Wolves
Art of Fighting
Ninja Masters
Waku Waku 7
Breakers (this was the most scandalous Street Fighter clone)
Rage of the dragons (a Double Dragon one-on-one fighting game)

3D Fighting

Sega were the first (as far as I know) to release a 3-D fighting game. In 1993 the released Virtua Fighter. Soon after other companies followed suit and released their 3-D fighting games. The best known besides Virtua Fighter are probably Tekken and Soul Blade (Soul Calibur) by Namco.

Virtua Fighter

Tekken

Although these games moved from 2-D images to 3-D the concept of the fighting games stayed the same.
Ironically Capcom themselves failed with their own 3-D Street Fighter game called Street Fighter EX.

The Here And Now

To this day Street Fighter is regarded as the King of Fighting games. Although fighting games are not as popular as they were in the 90's there are still a lot of fighting games being released. Series like Street Fighter, Tekken, Soul Calibur, Dead or Alive are still alive and kicking (pun intended).