The background is the scenery appearing to be behind the fighting action.
In fighting games, you rarely see much of any location besides what is in the background of a stage during fights, so many developers put a high amount detail into backgrounds.
Because 3D games have a high emphasis on graphics, they generally depict breathtaking locales, pretty buildings, phantasmagoric situations, animated crowds, and so forth.
In 2D games, too, you will find beautiful illustrations instead of rendered models (though in recent times, 3D backgrounds have also been done).
In games with characters from unrelated games as fighters, you will often see characters make cameos in the background, as well as more fun elements than more canonical-type games.
Sequels often feature re-made versions of older backgrounds.
When performing certain techniques like super special moves in some games, or especially long air combos in combo games, the background will change. With the latter example especially, this allows the background to have an arbitrarily tall/wide dimension.
Characters who come into the background of fighting game stories are often found literally in the background graphics. Some poor saps (like Tōdō Ryūhaku) end up being practically stuck in the background (he has only been a playable character so far in Art of Fighting, Capcom vs. SNK 2, and... well, some MUGEN stuff).
They can get promoted to flavor or secret characters if they are particularly well-treated by the developers or well-loved by fans, as well as downgrade the other way in the other case.
Sometimes, particular characters (like Geese in KOF) are popular enough that they are relegated to the background because there is a fear that they may be overwhelmingly popular in comparison to the other characters they are including in the game.