The back aerial (abbreviated as "bair", "BAir", or "ABA") is an aerial attack that is performed in midair by pressing the A Button, and tilting the control stick in the direction opposite to which the character is facing. Its official term varies between titles, where it is known as a "Back Midair Attack" in Super Smash Bros. Melee, and a "Back Air Attack" in Brawl. The majority of these attacks are kicks.
In general, back air in SSB64 are slow but with good range and good knockback. They are all 1 hit moves outside of Link
Pikachu: Kicks the opponent behind him. High priority move with good range, usually used either to end up tilt combos or to edgeguard the lesser priority recoveries like Fox (or Mario's down b for example), high power
Yoshi: Kicks opponent with both feet. Yoshi's overall slowest DJC aerial, good after up tilts, leads into other DJC aerials.
Samus: Raises her foot and then thrusts it behind her. Samus's overall best move. Huge range and power and Samus's main killing/spacing move.
Donkey Kong: Kicks opponent with both feet. Too slow to be overly useful, but good for comboing against wall
Link: Kicks feet behind him one at a time. Very quick move usually used for combos, more of a backwards double kick. Good in almost all circumstances and overall Link's most useful aerial for comboing
Luigi: Kicks opponent with both feet. Good range but slow, think Mario's but with more range and less speed
Mario: Similar to Luigi's. Good killing move, think Luigi's but with less range and more speed
Kirby: Kicks opponent with both feet. The best example of a wall of pain move of ssb64, good damage + good for combos, great edgeguard tool
Jigglypuff: Kicks opponent with one foot. An overall good move if it had more range, very quick, low power, virtually useless at low percents, can combo into itself at mid-percents
Fox: Kicks opponent with one foot. Yet another good aerial for fox, good for edgeguarding Falcon's and Samus's among others.
Captain Falcon: More of an advanced comboing move compared to his other aerials, more difficult to use in combos but overall better than up air against floaty character, or Samus, Luigi, and Jigglypuff. Good move to jump off platforms with.
Link: Two backward kicks. Very fast. Relatively weak, but surprisingly high priority and 1st kick hitting grounded opponents combos into several of his moves.
Zelda: The famous Lightning Kick. She jabs her heel behind her giving an extremely powerful initial, electricity-imbued attack for the first hitbox is sweetspotted. The second hitbox is very weak.
Sheik: Extends her leg backward very fast, the hitbox stays out long. The sweetspot is on her foot.
Ganondorf: Punches backward. Very similar to Captain Falcon's, but instead of having a long duration it has a large hitbox with a large sweetspot but fairly long landing lag.
Toon Link: Slashes backwards with his sword. It can be chained at low percents and be used as a kill move at higher percents. One of the best combos in the entire game with unbelievable priority. Even more priority than Mr. Game & Watch's Bair.
Samus: A backward kick. The sweetspot is on her foot, otherwise it's weak.
Zero Suit Samus: Identical to Samus', but slightly faster and stronger in terms of knockback.
Pit: Jabs his swords backwards. Good knockback, but hard to sweetspot.
Ice Climbers: Spin around with hammers slightly extended. Deceivingly strong.
R.O.B.: Pushes boosters behind himself. Moves him forward decent knockback.
Kirby: Extends both feet backwards. Used for Fence of Pain. Often considered the best move in Kirby's arsenal.
King Dedede: Similar to Kirby's, but with one foot. It has a large hitbox, with disjoint during the first frames of the move. Generally considered on of the best back aerials in the game.
Meta Knight: Slashes three times behind himself very quickly, slightly slower/more powerful than his forward aerial, which has the same properties.
Ivysaur: Spins around with vines extended. Low damage and very low knockback, but excellent range.
Charizard: Spins around surprisingly quickly with tail extended and wings slightly extended. 2 possible hits, one with wings and one with tail. Decent knockback on tail hit, wing hit will always link into tail hit. Has a sweetspot on the fire, but it's hard to sweetspot as it has a small size.
Lucario: Punches backwards. Aura extends from palm.
Jigglypuff: Spins around with foot extended to about twice its regular size. Used for the famous Wall of Pain.
Marth: Similar to Toon Link's in animation, Marth thrusts his sword upwards. However, this attack is notable because it turns him around.
Ike: A quick spin with sword extended. Powerful, can be auto canceled in short hops (it misses against smaller characters and very fast start-up but it has a very long cooldown like all of his aerials and moveset in general.
Ness: Plugs backwards with both heels. Initial hitbox quite powerful with electrical properties, second hitbox fairly weak, although buffed from Melee. Very Powerful.
Lucas: Flips and hooks his foot downward. Can spike at a certain angle (on the tip of Lucas's foot). Can kill at low %s.
Mr. Game and Watch: The famous snapping turtle, hits multiple times and has incredible priority, however it can be SDIed out of and be punished before the move has ended. Eats through shields.
Snake: Kicks backward with both feet, body becomes horizontal. Large duration and high knockback (although it deals the same damage, his body has slightly more knockback than his legs), but has a fairly long cooldown and 30 frames landing lag. Doesn't auto cancel in short hops. Very high priority and disjointed hitbox.
Sonic: Similar to Diddy Kong's, he spins around with his foot extended. One of his few KO moves.
R.O.B.'s back aerial can aid his recovery, although it does nothing to get closer when he's still flying after getting hit, as his hover pad lets out a quick burst, moving him forward quite alot. Also can hit enemys in front of R.O.B., both sides.
Kirby's back aerial in Melee, an attack similar to Jigglypuff's back aerial but instead has no start-up lag. The move is dubbed the "Fence of Pain".