Character Archetype is the defining term for a character's general strategy and gameplay style. In fighting games it is common to see certain recurring archetypes that compose and evolve the metagame.
Archetypes are universal: they can be applied to nearly every kind of game, be traditional board games, sports games, among others. The purpose is to inform the player what type of opponent they might encounter, and how to apply a plan of attack against said archetype.
There three categories of archetypes: offensive, defensive and flexible. Those, in turn, have different variations to specific kinds of strategy, and because of that, characters can fit into more than one archetype due their mindgames and gameplay style.
Offensive Archetypes
Offensive characters are the ones with better tools for approaching and chasing down opponents. They usually attack relentlessly without giving any breathing room.
All-Rounder
Also known as Balanced, they are usually easy to learn, with very balanced stats overall, being good at everything. Usually have both short range and long range attacks, as well as reliable KO moves, as well not having any major weaknesses. Because of that, however, they are usually the "jack of all trades, master of none" archetype, in that they don't really excel in anything, but are prepared for everything. With that in mind they can do very well in both offensive and defensive play. Examples: Mario, Yoshi, Pit/Dark Pit, Ryu.
Rushdown
Fast characters that specialize in close range combat and combos/juggling. Purely offensive, sometimes with no projectiles, but even if they use some long range attacks, they're always on the offense. Many of them are fast-fallers with good aerial speed. Examples: Fox, Sheik, Yoshi.
Pressurer
Characters that focus on close-range combat, usually to exploit openings and weaknesses on their enemies quickly and punish them with high damage combos, leaving no room for reactions. Examples: Pikachu, Peach, Roy, Kirby
Hit & Run
Fast melee characters that excel on getting in and out of their opponent range. While might not having many powerful attacks, with the few they have requiring good precision, they are quick enough to keep their distance and approach as fast as they retreat, punishing the mistake of their opponents. Examples: Captain Falcon, Greninja, Wii Fit Trainer.
Mix-Up
Perform best at close range, with versatile movesets based on tactics and mind games to exploit their opponent's weaknesses. They sometimes have with mid-long range projectiles that allow them to zone well, and are generally more well balanced than Pressure characters, however, not the the extent that All-Rounder characters are, since they can't play defensively, and have much more obvious/exploitable weaknesses. Examples: Falco, Lucas, Peach.
Zone-Breaker
These characters are formidable combatants as they have a bit of everything, combining Rushdown, Pressurer, Hit & Run and Mix-Up tactics. They can quickly disrupt opponents with their high speed and combos, completely dominating the battlefield while maintaining a safe approach. Examples: Sheik, Zero Suit Samus.
Dominating
These characters are similar to Rushdown and Pressure characters, but with very high power. They have very formidable speed, but also a damage output that rivals even powerhouses. They tend to have very high exploitable weaknesses, though, but are different from Glass Cannons as they do not suffer from low defenses. Examples: Roy, Cloud.
Footsies
These characters excel at close range ground combat, having superior priority and/or speed on their attacks while on ground. They tend to be less combo oriented that Rushdown, but boasting from superior power. Examples: Ike, Little Mac, Greninja, Ryu.
Glass Cannon
These characters have some of the most powerful attacks in the game, but at the expense of having very low defensive capabilities. They can dish out damage very fast and often have superior range. However, aside lower defenses, they also some kind of heavily exploitable weakness, such slow attack speed and/or low priority, being easily outmatched by characters that can outperform them. Examples: Zelda, Mewtwo.
Defensive Archetypes
The opposite of offensive characters, defensive characters usually avoid their opponents and bait them into setups to exploit their weaknesses from a distance before delivering a punishing blow.
Zoner
They focus on controlling the stage, mostly with tools and projectiles, or melee attacks that outrange opponents. They sometimes lack speed and power to fight, focusing on wearing down opponents from distance. Examples: Samus, Link, King Dedede, Marth.
Bait & Punish
Usually projectile users that pressure their opponents manipulate their behaviour, also having the tools for close range combat in order to take down opponent (though they do best from afar). They are usually equipped with projectiles/mid-long range killers. However, they normally face their opponent when attacking, unlike Trappers, who are more mind-game oriented. On the other hand, they can be purely melee oriented characters that use their superior speed and/or power to create openings and exploit opponents.
Alternatively, they can also be close or mid range characters that can pick off opponents slowly from a safe distance, exploiting their superior range to wear down (and annoy) their opponents. Mix-Up characters can also fall under here, as they also rely heavily on mind-games, however, they tend to favor a offensive fighting style over an defensive one. Examples: Bowser Jr., Captain Falcon, Wario.
Trapper
Usually also Zoners with some Bait & Punish tactics, those characters are utterly defensive, tending to avoid opponents (though not always) and lure them with traps that put a lot of pressure. They tend to have a lot of indirect options to constantly keep their opponents on their toes. They differ from Zoners in that they can attack while facing away and evading their opponents, and that they excel in multi-tasking, usually stringing combos with their traps and their close range moves. They're also much more difficult to use than Zoners as they rely heavily on keeping a lot of pressure and mind-games. Examples: Pac-Man, Robin.
Turtle
Most of the time these are also Zoners, being super defensive characters that focus on wearing their opponents down completely by hitting them from afar/with indirect moves. They can also be Hit & Run characters in the fashion that they poke opponents with projectiles and keeping safe distance. Some of these usually don't perform well at close range, but tend to have fast melee attacks. Examples: Link, Samus, Villager, Palutena.
Keep Away
Those can be considered the Hit & Run of the defensive category, but with some traits from Zoners. They excel in getting in and out of the opponent's space, not excelling in damage, but being very "annoying" as in keeping opponents at bay without letting them act or react. Examples: Samus, Sonic.
Stage Control
These can be considered a combination of Turtle, Zoner and Trapper, they excel keeping opponents at bay by having a variety of long range attacks to adapt every single situation and cover all their weak points, slowly building a barrage of projectiles that prevent opponents from approaching. Examples: Duck Hunt, Mega Man.
Tag Team
Also known as Puppeteer, those characters are basically two entities in one form or another. They have less range than most other Zoners, and mastering them is usually a very daunting, yet rewarding task. Generally suffer from low defenses, but not always. Examples: Ice Climbers, Rosalina & Luma.
Grappler
Grappling is usually a large part of their game, with many being also Bait & Punish specialists. There are no true grapplers in Smash Bros. due the different gameplay regarding it as a fighting game; instead all characters that have heavy focus on throws are considered Half-Grapplers. These characters are usually forced into defensive play because of their slow speed, large size and/or inferior jump (though not always), but make up for it with very powerful moves, and very little hit-stun. They essentially wait for their opponents to get in close to punish them for it. Examples: Donkey Kong, King Dedede.
Alternatively, Half-Grapplers can also be fast characters that not only have many setups and combos from grabs, but also good variety of moves to mix up their play; they're actually more offensive than defensive. While being quite agile, they lack the speed to be Rushdown characters, and the long-range to be All-Rounders or Mix-Up. Examples: Luigi, Ness.
Flexible
These characters can easily alternate between offensive and defensive gameplay as the situation demands, however, as they tend to incline towards another category, this can be considered a sub-category. This category also includes archetypes that are unique to a single character or that only fits in Smash Bros.
Dynamic
These characters are known for having a unique quirk or ability that affects their play strategy directly, Characters that rely on power ups, such as need to charge up a meter in order to use their stronger attacks or special stances that increases their parameters. They may have access to a variety of tools that allow them to effectively change their attributes. Examples: Shulk, Olimar.
Power
Exclusively to Lucario, this can also in a way relate to Dynamic as it revolves on a unique quirk. This refers to Lucario' s uniqueness of the Aura mechanic, where he deals more damage the higher his percentage is.
Precision
These characters rely on specific sweetspots in order to deal their full damage, or have attacks that requires pinpoint accuracy. If they hit with their sourspots, the attack power tends to be vastly inferior. Examples: Marth, Zelda.