If you had to classify a Dungeon Master/Player

If you had to define the types of these two, what would it be? Something like "The too loud player, the mary sue player, etc"

There are two types of players on this site. Chemgas650 and everyone else.

yeah, that about sums it up.
but to be real though, there are so many ways that people play, that there is no way to keep track of them.

Agreed with the last two posters, however a few stereotypes can be drawn to a lot of people:

On the subject of GM's
"The Inexperienced": GM's who have little experience in gm'ing or roleplaying overall and are very visibly clumsy in handling things. They are the ones that abuse railroading the most.
"The Torturer": Though i seen only a single person to ever fit this category here, these people are ones that make their tables brutal to everyone, and take joy in brutally ending characters' lives through the simplest of actions they make.
"The Storyteller": These are GM's who almost exclusively focus on the story and give little bother to everything else. You'll find combat in these people's tables to be a chore, and usually not very fun to do. It's also common for these people to host CC sessions way before they really host.
"The Combat Junkie": These GM's are the opposite of the storyteller stereotype. They give focus exclusively to combat, at the cost of the character's personalities. Their tables are mostly fun until you get to the story part.
"The Loremaster": These kinds of GM's have some of the best worlds and stories that they could host, but rarely do so, and when they do it, they can get rather finicky due to players getting their ideas completely wrong or going against concepts they have thought up for the world they made.
"The Multi-Host": GM's that have multiple tables that have varying degrees of effort put into them. They are quick to stop a table and start another if they feel like it. You'll usually find a bible of forums under their table with titles of their various tables.
"The Chemgas": ... No commentary.

On the subject of players:
"The Generic": Players who use generic kinds of characters with predictable personalities. They're usually beginners or inexperienced at roleplaying. They're not bad, everybody gotta start somewhere.
"The Low Effort": Players who don't really like writing more than two lines of text. Expect their actions and speeches to be as barren as their sheet and at max three lines of speech or action coming from their characters.
"The Bibliographer": People who like to write BIG actions and speech for the characters, spending way more time than an usual player does in what they write. Though they're fun with making endearing characters, they may clog up the pacing of the roleplay.
"The Silent": People who are silent both in and out of character. Not to be confused with The Low Effort, these players put effort into their sheet and sometimes in speech, but don't speak otherwise.
"The Debater": The kind of people to argue with the GM or everyone else over every single detail. Kick them if you will, or shut them down before they annoy you. Not really a fan of these people.
"The Injector": People who like to input their own ideas into whichever table they're playing. They mostly do it without the GM's allowance, often leading to either them being neglected it or forcing the GM into writing in a way so that their idea makes sense. Some of their ideas might be fun, but it can be hard dealing with them.
"The Eccentric": Players that are most likely to play characters that are eccentric in nature, having exaggerated traits to them in some way. They usually hog the spotlight.
"The Chemgas": ...No commentary.

Going to add one more to this. This is sort o a add-on for most of the above GM or Player classifications. A faucet of behavior really. Call me out on this if you have to as this is only from personal experience.

"The Reactionary": A player who scales their efforts into role-play based on involvement, actor input to immersion (player and GM), current flow of story, and their own literary skills.

While not really a classification of itself, this can sum up a player's involvement into the role-play. This is entirely opinion though. Do not take my word for it.

"The chemgas"
A combination of combat junkie, multi-host, and a tiny sprinkle of torturer when the GM.
A combination of the injector, the eccentric, and a dash of debater when playing.

(Speaking of debater i feel like that one's targeted at me)

holy %#@&, pale, you're alive?

^

damn pale

I sometimes use the idea from Magic: The Gathering, where there are 3 archetypes of players, Timmy, Johnny and Spike.

Timmy plays for fun. Only wants what is interesting, sees fun in the journey rather than the destination. Puts game as a social experience. Players who wants to do everything and enjoy the stories.

Johnny plays as a form of self-expression. Wants to show off to the world who they are. They like using specific fun strategies, mechanics or characters of their choice because they find beauty in it. Players who like to stick to one classes when character creating e.g. Warrior only, or mage only.

Spike sees the game itself as a competition, or an actual game, and likes to win. Whether it is PVE or PVP, they want to dominate it all, only concerned about quantity rather than quality. Results matter more than the process. These players like to use the strongest class, by analyzing core rules, specific table rules, GM's habits, tactics, learning the math behind it all, and getting the best loot.

There are pros and cons in each archetype, but generally different minds have different views about each of them.

-- digs this buried post and shoves it up above ground --
Hey I just want to add one more to classification of GMs, don't mind me!

"The Fanboi": When one has divulged enough in other fiction, whether well-known or not and makes their table entirely based on them. This is likely to attract others of same fandom as "The Fanboi" or at least players who show the slightest of interest in them. You can say that they are a type of "The Loremaster" except that when the GM makes a mistake, it is so easy to point that out, especially from a more knowledgeable "The Debater" player(IT is not the GM's own world after all!). "The Injector" player are also highly discouraged from playing at this table, more so if they know nothing or little about the fandom.

I'll give you some examples of some of the more common fandoms I've seen so far:- Star Wars, Dragon Ball, Naruto, JoJo's Bizzare Adventure, Fire Emblem.

Other non-original fiction can have great potential for roleplay, and I personally encourage this. At least we're having fun right?