A buff is a term used to describe the improvement of a character or a specific aspect of a character, either through updating a game or by releasing a sequel. The opposite term is to "nerf" a character.
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. Another example is Samus from the original to Melee, where she had overall one of the game's worst moves to one of Melee's best moves, and is also a faster character overall.
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.
Has a new Special Move: Raptor Boost. His air speed was increased significantly (went from the 3rd slowest to the 3rd fastest, making him a faster character). Was given a new F-air, the Knee Smash, a devastating horizontal finisher. His old F-air was made his N-air and has more vertical range. All of his Smash Attacks were changed and made more powerful (He has the 3rd most powerful F-Smash and the second strongest D-Smash. His U-Smash is no longer the weakest, now being among one of the strongest in Melee) with two of them with more reach (particularly U-Smash). Falcon Dive has more distance, horizontally and vertically. D-tilt has better range and combo ability. Falcon Kick now restores his midair jump if previously used, improving his recovery greatly. U-tilt is slightly stronger (despite it no longer being the strongest and hitting only with the sourspot), Rapid combo is faster and has bigger hitboxes, giving it more reach, has a new F-tilt with better range and power, has faster running and walking (to compare his speed with Melee's fast-paced gameplay, and also, his running and walking animation were changed significantly), and finally, Falcon Punch is much stronger. A much stronger character overall with much better recovery and speed.
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. His blaster can now rack up greater amounts of damage, as it now has rapid-fire capabilities. Fire Fox goes almost twice the distance it previously did, and can now KO.
Has a new Special Move and KO option: Rollout. Rest is much stronger, Pound is much faster with better Priority, all jumps now give height, increased reach in its attacks, its air speed is faster, its dash attack is now KO potential under 125% (near the edge). Overall a stronger and faster character.
Has a new Special Move: Hammer Swing. Forward tilt is stronger, can now Kirbycide. Also his Inhale 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. It is also claimed to be his best finisher, besides his F-Smash, U-Smash, his front and back throws, and his Swallow move. However, these good buffs don't prevent him from being severely nerfed.
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 faster two hits, replacing the one slower slash. He is the third most buffed character (Behind Luigi and Samus).
Has a new Special Move: Green Missile. Has a new F-air with much more knockback. His D-air 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 much more knockback. He's the second most buffed character overall behind only Samus. Due to his attacks being mostly the same as Mario's, he is a faster character.
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 much faster. He has a new F-Smash that is much stronger. Faster overall.
He now has a very strong projectile in PK Flash, a new Special Move. B-Air is slightly stronger. He is also slightly heavier compared to N64 (he was lighter than Fox, now he's much heavier than Fox, weighing 19 units more in NTSC and 21 units more in PAL, because Fox is slightly lighter in this version). F-Smash is stronger (when tipped) and has more range.
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. It has more powerful attacks overall.
Her recovery was improved significantly, has a new very effective projectile in the Missile, a new Special Move. U-Tilt is much faster and better for comboing and KOing; Her N-Air (Sex Kick) is made stronger and is the second strongest sex kick. Has a new D-tilt that is much stronger. She is also a faster character overall. She was the most buffed character from the original.
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.
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, 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. F-Smash hits twice to rack up 14% more. Has an increased running and walking speed. Faster and Stronger overall.
Falcon Punch is faster and can be reversed for the almost the same power as his Melee counterpart. 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, has a bigger grab-zone (similar to his in SSB), and is slightly stronger (in terms of damage, 15% to 17%). Jab, Up tilt, Down smash, and U-air received very minor buffs (the former two have slightly better range and the latter two are slightly faster).
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.
U-Tilt, U-Smash, and U-Air are all 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.
D-air is drastically stronger (can now KO). D-Smash is slightly stronger and has better range. Air speed was increased (went from the slowest along with Falco in Melee, to somewhat below average in Brawl, similar to SSB, but slower). U-Smash has more vertical range (similar to Falco's in Melee), combined with the ability to perform a smash out of a dash, makes it much easier to land. F-air is faster, has weaker knockback except the last kick (allowing Fox to land all kicks on an opponent) and it gives Fox vertical distance, giving him an additional recovery option. Reflector now has knockback scaling, improving semi-spikes at higher damages. Neutral combo is much faster, making it better for racking up damage.
Dash attack now has more power, speed, range and can KO under 125% damage (Second strongest dash attack in the game, only exceeded by King Dedede's). F-Smash now has signicantly more horizontal knockback and deals slightly more damage. His U-Air is stronger. Jab is stronger in terms of knockback, albeit significantly slower. U-Smash can now hit foes on the ground. U-Tilt and Warlock Punch charges faster with the latter also being able to reversed. D-Smash is easier to trap and is faster (In terms of start-up). D-Tilt is much stronger. His Gerudo Dragon was replaced with Flame Choke making it a better Self-destruct method and is better at lowering character's defenses. F-Tilt is stronger. His Dark Dive has an added effect to it, if he fails to grab a foe and it can now grab foes on a ledge or hit them off, making him a much harder character to edgehog. Strangely, Ganondorf's buffs are very similar to Captain Falcon's.
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.
Hero's Bow is faster and travels much further. Clawshot finds and tracks the walls, making it easier to use it as a tether recovery. It also has less ending lag. The first hit of his F-Smash is much stronger. His U-Smash is much easier to connect due to having a much larger hitbox. D-Smash is stronger. Spin Attack now charges up for more power.
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.
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. His U-Air and B-Air are stronger. Has a new Down special, F.L.U.D.D.
Dolphin Slash improved. Counter damage now scales with the countered attack and the counter frames start sooner. Dancing Blade is faster and easier to perform. His tipped attacks are generally stronger and with much higher base knockback.
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.
D-Air has better range and power, Dash Grab has better range and is faster, his pummels are much faster and much better at racking up damage. His Grab range was increased (no longer the second shortest). 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. His F-Smash is stronger (went from below-average to above-average). D-tilt is much faster and much better at racking up damage on foes (it causes high damage for a tilt, 4%, and for its speed). Stronger and Faster overall.
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).
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. The second hit of her Forward Smash has increased knockback, making it more reliable in battle.
Egg Toss gives a slight vertical and horizontal boost that can be used for recovery, several tilts and aerials improved (His U-Air is much stronger, his F-Tilt is slightly stronger, due to Brawl making short hops much easier, his D-Air is more useful in battle), faster air speed (he is now the fastest character in mid-air of the game). His F-Smash causes 1% more damage. Like Bowser`s Bowser Bomb, Yoshi Bomb now hits during the rising animation and causes 5% more damage (19%-20% total). Despite Yoshi having a shorter tail from Melee, he has more reach in any attack involving his tail.
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 and faster overall.