A stage (sometimes “level”), in a fighting game, is the place where you spar with your opponent.
Depending on the game, a stage may be representative of a particular character.
The stage has a particular background associated with it, and may have its own background music (though sometimes the music goes along with whatever character is on).
Stage sizes may change, and in 3D games especially, stages will have different arrangements, such that some will have the edge of the ring and the walls in different places, which certainly changes up the strategies you must use. Some games let players choose stages, too.
Some stages have various obstructions, and beating on your opponent against these often adds extra damage. Particularly in the ultra-gorey fighting games, these may cause appallingly violent deaths. Things can break, too (like in Street Fighter II, Fatal Fury 2, and Real Bout Fatal Fury), and sometimes this reveals new scenery (like in the Mortal Kombat and Dead or Aliveseries, or Kaiser Knuckle and the Dankūga prototype).
Many stages are very much modeled after real-world locations, though just as landmarks are in movies, there may have been liberties taken with the actual layout of things.