There are 4 main categories for a character’s archetype - Regular, Zoner, Grappler, and Dynamic. There are all rounder characters but they can lean towards one group more so the category is more or less a 5th.
Most fighting game characters in existence can fit here. They fight with their physical bodies and don't rely on toying with the opponent. They get the kill as fast they can. The most basic type and you can see it in older fighting games most because of the limitations.
Rushdown - They get in close and their sweet spots are closer, most of the time they are easy to learn and are powerful characters. (Examples: Cammy, Fei Long, and Rufus from Street Fighter; Jam from Guilty Gear; many characters in King of Fighters)
Mix Up - They get in close but rely on forcing opponent into tough situations and guessing game. (Examples: El Fuerte, C Viper, Urien, Makoto from the SF series)
Dominating - They have great normals and specials, which give them the power to do anything. Most of the time they simply have high damage output one way or another. The key factor is that they are designed to be powerful and high tier most of the time. (Examples: Sagat, Yun from the SF series; Chang and Maxima from KoF)
They have all the tools, and are good at everything. Different than simply combining two types. This is a weird section because it is subjective. (Examples: Ryu, Bison, Chun Li, from SF.)
Their projectiles are a big part of their game, or, they have tools to control the fight rather than win with strength and speed. They do not fight on their own and that's their defining feature.
Trapper - They have projectiles that control the opponent's space rather than deal straight up damage. A subset is the physical trapper (Dhalsim from Street Fighter, Billy Kane from The King of Fighters). Most turtles (non-rushdown characters) can fall in this category. (Examples: Testament, Dizzy and Venom from Guilty Gear; Sagat, Guile, actually most projectile users in Street Fighter; Strider in Marvel 2)
Puppet/Tag team - Characters that are actually 2 different entities. Usually the player controls one and the second is a puppet under your control. This is different from trappers in that they usually have less range and rely more on close range zoning and is more like playing two characters at once. They also have higher execution to play well usually. (Examples: D'Bo from Jojo's BA; Liselott from Arcana Heart 3; Carl from Blazblue.)
Their command throws are a big part of their game. There are very few pure grapplers. Most have other types thrown in like Dominator and Mix Up.
Straight-up Grappler - They are slower, stronger, and always want to get in. Straightforward and basic type. The grapple is their main game.
Half-Grapplers - Take another archetype and give them a few grabs. This is a pretty big deal in some games, and not so in others where half the cast has a command throw. The difference between this and other grapplers are that they are considerably faster, throws are more mobile and not just basic SPD/Falling Sky. (Examples: Blue Mary and Shermie from KoF, Abel, Rainbow Mika, and Alex from the Street Fighter series. She Hulk and Hulk from Marvel can be argued. Marvel vs Capcom 3 as a game doesn't make room for pure grapplers.)
They change throughout the fight, which affects their style. This can be user controlled, or not. There are no pure dynamic types, they always mix in with others.
Stances - They have multiple stances which they can access at any time, so they can be played in a lot of ways. This may not seem important but stance based characters deserve their own section since it's one of their defining features. (Examples: Gen from the Street Fighter series; Jhun, Chin, and Vanessa, to an extent, from King of Fighters)
Power-up - They can power themselves up, but usually they're vulnerable so choosing when to power up is necessary. A good player can play totally different at the end of a round compared to the first few seconds. This can include characters with stances, but stances that transition into each other in a line rather than swapping on the fly. Think drunken boxing guy from Tekken, it's a stance but you work to get there. That's the key difference. (Examples: Jam from GG; Hakan from SF; Maharia from Jojo's BA; Frank West in Marvel 3)
Ammo characters - For whatever reason, the characters’ moves are limited and they must replenish to keep using them. They are usually powerful and this mechanic makes you think about your next move. (Examples: Petra from Arcana Heart 3; Robo-ky from Guilty Gear)