Background characters are, as the name implies, characters that appear in the background of some stages, they will appear in a stage that correspond to their home universe (i.e. Shy Guys appear in Yoshi's Island). Note that background characters shouldn't be confused with stage hazards.
Necky appears flying in the background.
King Dedede and Bronto Burt appear flying in the background.
King Dedede as a background character in dream land
Several enemy sprites ripped directly from Super Metroid, including Ridley, can be seen flying around.
Several flying- and bird-type Pokémon, including but not limited to Pidgey, Fearow, Moltres, and Butterfree, fly across the skyline.
Shy Guys appear in the background.
A Goonie, in its Yoshi's Story design, occasionally flies in the background.
Sometimes, Arwings are seen flying in the background.
A Chozo statue on the main platform comes to life and briefly walks before settling down into its original position.
Mother Brain can be seen in the far background.
Kraid emerges from the lava and swipes at the stage, turning it clockwise or counter-clockwise.
A U.F.O. which can be boarded hovers in close proximity to the Monotoli building.
Porky Minch appears in his helicopter.
After three minutes have passed, the Four Giants appear on the hillside to push the falling moon back into the sky (This is in reference to the fact that Link had three days to wake said giants and prevent the Moon from falling).
Cranky Kong is seen walking back and forth in his cottage and rocking in his chair at various times.
A Birdo moves in from either the left or right and shoots eggs into the battlefield and a Pidgit flies in on its magic carpet. Only the Birdo can be KO'd.
Birdo in Kingdom II
The meteor from the beginning of EarthBound can be seen pulsating at the top of the hill, while one of the car hazards is the Runaway 5 tour bus.
Numerous giant floats depicting a variety of Pokémon appear, including Squirtle, Onix, Psyduck, Chikorita, Weezing, Slowpoke, Porygon, Wooper, Sudowoodo, Snorlax, Venusaur, Seel, Wobbuffet, Unown, Goldeen, Lickitung, Chansey, and Geodude.
A Bulborb sometimes appears to the right as part of the stage, and also in the background.
Tails, Knuckles, and Silver can be seen running up and down the shuttle loop.
Tails, Silver, and Knuckles.
Though not characters as such, several weapons of the ship from appear and attack players.
Kirby Super Star
Boos briefly appear from the rubble of the mansion when it collapses.
King of Red Lions sails behind the pirate ship as a small platform.
A watchtower, or Sea Platform, that is occasionally passed fires on the pirate ship from the background.
Just as in the original F-Zero GX, giant cranes shaped like R.O.B. appear on the background of the stage.
Dialga, Palkia, and Cresselia appear on occasion in the background of the stage. Their attacks can affect gameplay or alter the terrain of the stage.
Uxie, Azelf, and Mesprit may occasionally appear on the stage at random.
For a third time, Shy Guys appear in the background, with Chain Chomps also occasionally appearing.