A buff is a term used to describe the improvement of a character or an specific aspect of a character, either through updating a game or by releasing a sequel. The opposite term is "Nerf".
A good example of a character being buffed is Zelda in Brawl, where she has quicker, stronger attacks, in strong contrast to her attacks in Melee. Another example is Kirby, who has completely regained his power and has increased speed.
Characters who had attributes buffed from SSB to SSBM
There are unique buffs from SSB. Characters have the ability to air dodge and step-dodge. Some might consider chargeable Smash Attacks is a buff. Also the game has faster gameplay, which helps character survivability. Most characters have new Special Moves in the form of Side-B moves, while some characters, like Link, had their Neutral-B moved to their Side-B.
Captain Falcon
Has a new Special Move: Raptor Boost. His air speed was increased. Was given a new F-air, the Knee Smash, a devastating horizontal finisher. His old F-air was made his N-air. All of his Smash Attacks were changed and made more powerful. D-tilt has better range and combo ability. Falcon Kick now restores his midair jump if previously used. Stronger overall.
Donkey Kong
Has a new Special Move: Headbutt. His Giant Punch is faster and has less ending lag, his air speed is much faster, and he is a faster character overall. Has a new U-air that can combo opponents. All Smash Attacks, particularly D-Smash and F-Smash, are slightly stronger.
Fox
He now has the ability to infinite shine, wave shine, and shine jump cancel. He has faster movement and attacks, his attacks are stronger, and he has an additional recovery option with Fox Illusion, a new Special Move.
Jigglypuff
Has a new Special Move: Rollout. Rest is much stronger, Pound is much faster, all jumps now give height, increased reach in its attacks, its air speed is faster, its dash attack is now KO potential under 125%. Overall a stronger and faster character.
Kirby
Has a new Special Move: Hammer Swing. Forward tilt is stronger, can now Kirbycide. Also his swallow buffed due to the better self destruct method and having more characters to choose from. Has a new U-Air with much more knockback than his previous one. His U-Air in Smash 64 was moved to his N-Air in Melee, therefore replace the sex kick he had in N64. It is also claimed to be his best finisher, besides his F-Smash.
Link
Has a new Special Move: Hero's Bow. His recovery is much better (from arguably the worst to one of the best). He is a faster character overall. Better SHFFL so he can combo easier and his air game is improved. He also has a new F-Smash of two hits, replacing the one slash.
Luigi
Has a new Special Move: Green Missile. Has a new F-air with much more knockback. His dair has also changed slightly, with increased knockback and it can now Meteor Smash like Mario's new F-air. He also has a new F-Smash with slightly more knockback.
Mario
Has a new F-air, which has the ability to Meteor Smash. He has a reflector in his Cape, a new Special Move. He has also gained the ability to chain throw opponents. His U-Smash and D-Smash are faster. He has a new F-Smash that is slightly stronger.
Ness
He now has a very strong projectile in PK Flash, a new Special Move. B-Air is slightly stronger. F-Smash is slightly stronger even when uncharged.
Pikachu
Its D-Smash can now combo opponents. It has a new special in Skull Bash that can be a powerful KO move as well as aid in recovery.
Samus
Her recovery was improved, has a new very effective projectile in the Missile, a new Special Move. She is also a faster character overall.
Yoshi
Has a new Special Move: Egg Roll. His recovery was improved. Also his D-Smash is faster and more powerful and having no starting lag at all, making it one of the fastest D-Smashes in the game.
Characters who had attributes buffed from SSBM to SSBB
There was an overall buff to shielding and dodging: faster shields (both in putting up and dropping), longer perfect shield window, faster sidesteps (around half with additional multiple IASAs), faster rolls, and infinite air dodges. This, combined with an overall reduction of hitstun, has made Brawl a much more defense based game than its predecessors. All veterans have better air speed, most notably Fox.
Bowser
Fire Breath now can be aimed up and down, recharges faster, and has longer range. F-Tilt, F-air, U-tilt and Jab all have much larger hitboxes, and consequently, much better range in all respects, with the latter three also being disjointed. Bowser Bomb falls faster, is much stronger (can reliably KO under 100%), and also hits during the rising animation when used on the ground, jumps higher when using U-smash, giving him better vertical reach, increased jump height, increased air speed. Faster overall.
Captain Falcon
Falcon Punch is faster. Raptor Boost now is easier to use for both attacking and recovery. Falcon Dive can grab foes on a ledge, making him harder to edgehog. Jab, Up tilt, Down smash, and U-air received very minor buffs (the first two have slightly better range and the last two are slightly faster).
Donkey Kong
Fully charged Giant Punch now has super armor frames. Headbutt is faster and stronger. D-Tilt and F-Tilt when angled down now trip. Spinning Kong travels slightly farther, both horizontally and vertically and has super armor frames when used on the ground. U-Smash can hit grounded opponents. F-Smash and D-Smash have much higher knockback. Hand Slap has more range and damage. He has better approach options and increased shield size. Several other attacks have slightly increased range, speed, and/or power, making him an overall better character.
Ice Climbers
Both can grab ledges simultaneously. Belay is a Tether Recovery if used near the ledge. Lone Belay gives slight distance. The partner's AI is increased.
Falco
U-smash and U-air are much stronger. Falco Phantasm is faster and travels farther, making it a better attack and recovery move. F-tilt has better range. Several attacks were changed and given more range and disjointedness, such as jab, F-smash and his Reflector. N-air is better at setups. Can now SHDL. Down throw now chain throws and is one of the best chain-throws in the game due to its versatility. He is also less vulnerable to combos and chain-throws. He is perhaps, the fast-faller least vulnerable to chain-throws.
Ganondorf
Dash attack now has more power and can KO under 125% damage. F-Smash now has signicantly more horizontal knockback and deals slightly more damage. Jab is stronger in terms of knockback, albeit signifcantly slower.
Kirby
Improved throws, increased priority, much stronger smashes (nearly as powerful as his first incarnation), slightly faster air speed, yet still slow. Swallow can eat items and has increased range. Throws are now inescapable. Faster and much more powerful overall. He is on par with Mr.G&W for being the most buffed character in Brawl.
Luigi
F-Smash has much higher knockback. B-Throw is stronger. Green Missile can be charged infinitely, and misfire travels a shorter distance to prevent SDs. Vertical momentum is carried into Green Missile and Luigi Cyclone, making it much easier to move vertically. Super Jump Punch is more like its SSB counterpart, and covers more horizontal distance when helpless; lateral air speed is generally faster but is still the second-slowest. Down Taunt has higher set knockback. Stronger overall.
Mario
Super Jump Punch has more knockback. Has a new D-air, the Mario Tornado, which is faster and has more knockback than his Melee D-air. Cape carries any momentum with it, allowing for new techniques such as the Cape Glide, and reverses foes' momentum. His U-smash has more range.
Marth
Dolphin Slash improved. Counter damage now scales with the countered attack. Dancing Blade is faster and easier to perform. His tipped attacks are generally stronger and with much higher base knockback.
Mr. Game & Watch
Increased priority, larger shield, Fire now produces a parachute and he can attack after using it. Has a new N-Air. Oil Panic can cancel all knockback dealt to him. D-Smash has a larger sweetspot and stronger sourspot. Much better range with D-tilt. Even though his overall speed didn't change, Brawl's gameplay is slower, putting him at less disadvantage than Melee. He is on par with Kirby for being the most buffed character in Brawl.
Ness
D-Air has better range and power, Dash Grab has better range and is faster. PK Thunder 2 has less lag and is much more powerful, U-Smash and D-Smash now have KO potential. PK Flash charges slightly faster and is easier to control. PK Fire is faster, B-Throw is slightly more powerful. Stronger overall.
Pikachu
Thunder has less ending lag and is stronger when it connects with Pikachu. Has increased grab range, no longer having the shortest. Quick Attack can now be used to perform a QAC. D-smash is much better at KOing and racking up damage. He is also capable of performing large chain-throws (he is one of the four premier chain-throwers in Brawl).
Sheik
F-tilt can now lock most characters. Chain moves more freely and can be used as a tether recovery. Vanish has less lag and is stronger in terms on knockback. Needle Storm now stops charging when it is fully charged instead of having to save them manually.
Yoshi
Egg Toss gives a slight vertical and horizontal boost that can be used for recovery, several tilts and aerials improved, faster air speed (he is now the fastest character in mid-air of the game).
Zelda
Din's Fire now has a larger hitbox, is much faster and stronger, and has less ending lag. Farore's Wind creates an explosion upon reappearing. U-Air is a much better finisher, sweetspotted aerials have increased in knockback, slightly increased dash speed and a faster grab. Stronger overall.