Weight is a measurement of how much a character can resist knockback. Weight is one of several factors used in calculating the amount of knockback a character experiences. Characters with a higher weight (heavy) tend to suffer less knockback, and characters with a lower weight (light) tend to suffer more knockback, all other factors controlled.
Weight is, in practice, understood as how difficult it is to send a character flying away. Because of this, it is generally considered an advantage for a character to be heavy, as less knockback makes it harder to KO a character. Additionally, in Melee, Brawl, and Smash 4, many throws take longer to execute on heavier characters, giving the opponent being thrown more time to properly react to throws and to DI effectively. Because throwing heavier characters causes the throwing animation to continue longer once the target is released, the viability of throw combos may be affected; for example, in Melee, Captain Falcon is not vulnerable to many of the up throw chain grabs (such as from Marth) that the space animals are vulnerable to; in Brawl, the Ice Climbers' infinite chain grabs are more difficult to perform on heavyweights; and in Smash 4, Robin's down throw to up aerial "Checkmate" KO setup is the hardest to perform against Bowser and Charizard, since they are heavy but have low falling speeds. In Ultimate, weight is a factor in determining how much a character can be pushed by an opponent running into them, and how far they will push other opponents.
Super Smash Bros. characters are often divided into separate "weight classes", with the most common method being a separation of the roster into three roughly equal groups: lightweight, middleweight, and heavyweight. Weight class divisions are often arbitrary and not universally agreed upon. "Middleweights" are usually described as characters with a close to average weight, or a similar weight to Mario, who is often seen as the "default" character in Super Smash Bros. In Melee, "lightweight" usually encompasses characters who are knocked down by Fox's shine (with a weight of 85 or below); thus Marth is often considered a middleweight in NTSC but a lightweight in PAL, due to his decreased weight in that version of the game. In Brawl "lightweight" often encompasses characters who are knocked down by King Dedede's down throw, with a weight of 85 or below (Zelda, Sheik, and all lighter characters).
Another commonly used weight class is the "super heavyweights", which describes the heaviest characters in the series, who also usually have the largest sizes and hurtboxes and often shake the screen upon landing on the ground. These characters commonly include Donkey Kong, Bowser, King Dedede, Charizard, King K. Rool, Incineroar, Snake (Brawl only), and Ganondorf (Ultimate only).
Lighter characters do have their own advantages. Because of the higher knockback they receive, they can usually escape combos earlier. However, since weight affects high knockback values more than low ones, this advantage is less significant compared to the disadvantage of being easier to KO; the size of a character's hurtboxes, as well as their falling speed and gravity prior to Ultimate (and especially in Melee), have a larger impact than weight on how vulnerable a character is to combos. As previously mentioned, a few specific combos are less effective on lightweights to a much greater degree, such as Fox's waveshine combos in Melee and King Dedede's down throw chain grab in Brawl; in both cases, characters with a weight value below 86 receive enough knockback to the point where they are put into tumble, allowing them to tech or execute a floor recovery and escape subsequent uses of such moves, whereas heavier characters are stuck in their non tumble animation, making them very vulenrable. Other advantages to light weight include weight-sensitive platforms, such as those in Mushroom Kingdom and Rainbow Cruise, which fall slower while holding less weight, making them safer to use for lighter characters. Finally, certain weight-based throws have hitboxes that appear for very short lengths of time, and if the target is very light, the animation may progress so quickly that the hitbox appears and vanishes in less than a frame without hitting — this causes the lightest characters to take less damage, notably with Bowser's down throw in NTSC Melee and Link's down throw in Brawl, both of which miss their pre-throw hit on Jigglypuff (and Mr. Game & Watch in Bowser's case). Because of this, most throws with hitboxes from Brawl onwards are not weight dependent, although there are a few exceptions, such as Link's aforementioned down throw.
Heavier characters tend to have stronger attacks, longer range, worse recovery (that is; recoveries that are slower or cover less distance), bigger hurtboxes, and slower movement, while lighter characters tend to have weaker attacks, shorter range, better recovery, smaller hurtboxes, and faster movement. However, this is a loose trend that many characters defy in multiple ways — Captain Falcon is heavy but dashes incredibly quickly, Yoshi is heavy but has a high jump and weaker attacks, Wario is heavy but short and highly maneuverable especially in the air, Zelda is light but loaded with powerful attacks, Falco is light but has a very high falling speed, and Little Mac is light and speedy with excellent frame data but has among the worst recoveries in the games he appears in, which is the exact opposite of King K. Rool, who is heavy and slow with poor frame data but has a great recovery.
In single-player modes, sometimes unnaturally high weight is introduced to challenge the player, such as when fighting Metal Mario; this is often paired with additional knockback resistance, since even characters with infinite weight will still take knockback from any attack with a base knockback greater than 0.
On an additional note, some attacks are weight-independent, and will always act as if the target fighter's weight is set to 100; the amount of knockback dealt will remain consistent, no matter how light or heavy the target is. This is especially notable for attacks that have the bury, paralyze, or stun effects. However, other factors, such as knockback-taken multipliers, and knockback resistance, still affect the amount of knockback dealt.
Super Smash Bros. weight values
These values use a different scale than the later games - heavier characters have lower numbers, representing a direct multiplier in the knockback formula. For comparison purposes, the equivalent value in the newer system is also listed. The conversion rate from the old system to the newer system is (200/weight) - 100. 
Super Smash Bros. Melee weight values
Following are the characters in Super Smash Bros. Melee, ranked in order of heaviest to lightest. The Metal Box multiplies a character's base weight by 3, the Super Mushroom by 1.6, and the Poison Mushroom by 0.625.
The average weight value in NTSC version for all fighters is 90, meaning that Peach, Sheik, and Zelda are the characters whose weights are the average.
Super Smash Bros. Brawl weight values
This is a list of characters' weights in Super Smash Bros. Brawl.
The average weight value for all fighters is rounded down to 94.79, meaning that Sonic's weight is closest to the average.
Super Smash Bros. 4 weight values
This is a list of characters' weights in Super Smash Bros. 4.
According to the game, weight is the base factor in determining how many powers can be equipped in Smash Run. There are other factors — namely a character's speed — so it is not a direct linear correlation, but as a general rule weight definitely carries a positive correlation: the higher the weight, the greater the number of powers, with the only exceptions being Miis and the DLC fighters having a power limit of 25.
Certain updates have made slight alterations to the weights of characters, affecting both survivability and resistance to combos. These changes have not had a major effect on most characters in the metagame, with the exceptions of Sheik and Bowser, who are considered to be worse and better after their weight alterations, respectively.
The average weight value for all fighters (except Miis) is rounded down to 94.76, meaning that Robin, Pac-Man, and Roy are the characters whose weights are closest to the average.
Super Smash Bros. Ultimate weight values
This is a list of characters' weights in Super Smash Bros. Ultimate.
Mii Fighters now have their weight determined by their class rather than their size modifiers from their creation.
The average weight value of all fighters is rounded down to 95.87, meaning that Pit, Dark Pit, Ivysaur, and Wii Fit Trainer are the characters whose weights are closest to the average.