Alright boys and girls, pull up a chair! It's history time!
Note: Though most of this is my own knowledge, I will be supplementing with some notes from the Fire Emblem Wiki section on archetypes. Won't be touching on FE's 2-5 as I am not familiar enough with them!
The Jeigan/Jagen
Two spellings of the same name, these guys are the bane of every new player. They are the old ****s. That's almost always the excuse for why they're bad, is being old or sick.
They're pre-promotes (almost always paladins) given to you at the start of the game. Compared to your other units, they are gods among men, having bases far higher than your unpromoted units. However, they are still usually pretty bad, and have HORRIBLE growths. Using one leads to an underleveled army deprived of EXP, and eventually the game catches up to the Jagen. An inexperienced player may find themselves in a position where their Jagen is one of few passable units, and has a hard time completing the game.
This happened to a lot of people with Marcus in FE7, though he's not technically a REAL Jagen. He just can't do the whole game himself. (Almost all of it, though)
Notable Jagens: Jagen himself (SD), Old Marcus (6)
The Oifey
Introduced in FE4 with Oifey himself, Oifeys are very closely related to Jagens, except they DON'T SUCK. They're Jagens with better growths, so they're not useless lategame.
Notable Oifeys: Marcus (7), Seth (8), Titania (9), Sothe (10), Frederick (Awakening)
The Est
As I mentioned before, characterized by late join times, high growths, and awful bases! Usually turn out to be some of the best units if used.
Notable Ests: Sophia/Zeiss (6), Nino (7), Ewan (8), Elincia (9), Pelleas/Kurth (10)
Cain and Abel
These guys are your cavalier pair that we've had since the dawn of time. Red one usually focuses on Strength, Green on Speed. Sometimes backwards, sometimes not the right colors at all! Usually get em very early, exception being mid-game Kieran.
Cain and Abel pairs: Cain/Abel (SD), Alan/Lance (6), Sain/Kent (7), Kyle/Forde (8), Kieran/Oscar (9/10), Sully/Stahl (Awakening)
The Ogma
9/10 times a Mercenary, Ogma's are characterized by high bases and above average growths, resulting in a very good unit overall.
Notable Ogmas: Ogma (SD), Dieck (6), Raven (7), Gerik (8), Ike (10), Gregor (Awakening)
Bord And Cord
Fighters in pairs! Again with the Strength vs. Speed thing.
Notable Bord/Cord pairs: Bord/Cord (SD), Wade/Lot (6), Dorcas/Bartre (7), Ross/Garcia (8), Boyd/Nolan (10)
The Nabarl/Navarre
Again two spellings. This archetype is the Myrmidon you recruit from the enemy (usually) early in the game. Usually related to the Lena/Julian archetypes via plot.
Notable Nabarls: Nabarl/Navarre (SD), Rutger/Fir (6), Guy (7), Joshua/Marisa (8), Zihark (9/10), Lon'Qu (Awakening)
The Gotoh
The god-mode unit you get at the end of the game to crutch your way through.
Notable Gotohs: Gotoh (SD), Karel (6), Athos (7), Pretty much any Laguz Royal (9/10), Nagi (SD), Basilio/Flavia (Awakening)
The Lena
Your early cleric! Good magic growths, usually related to Julian/Nabarl. Almost always female, exception is Rhys.
Notable Lenas: Lena (SD), Elen (6), Serra (7), Natasha (8), Rhys (9), Laura (10), Lissa (Awakening)
The Julian
Early thief! Usually related to Lena/Nabarl.
Notable Julians: Julian (SD), Chad (6), Matthew (7), Colm (8), Volke/Sothe (9), Sothe/Heather (10 - He's both a Jeigan and a Julian in RD), Gaius (Awakening)
The Pegasus Sisters
Three siblings, usually Pegasus Knights, have the ability to do the Triangle attack once it's introduced.
Peg Sisters/Trios: Est/Palla/Catria (SD), Yuno/Thany/Tate (6), Wendy/Bors/Barth (6), Florina/Fiora/Farina (7), Tana/Vanessa/Syrene (8), Marcia/Tanith/Elincia (9), Oscar/Boyd/Rolf (9), Sigrun/Marcia/Tanith/Elincia (10), Barst/Bord/Cord (SD)
The Maria
The Cleric/Troubadour usually low level, who's been held hostage or introduced after Lena in the plot.
Notable Marias: Maria (SD), Clarine (6), Priscilla (7), L'Arachel (8), Mist (9/10), Maribelle (Awakening)
The Gordin
Your generic level 1 Archer!
Notable Gordins: Gordin/Norne (SD), Wolt (6), Will/Rebecca (7), Neimi (8), Rolf (9), Leonardo (10), Virion (Awakening)
The Draug
Standard Issue Armor Knight MK.5
Notable Draughs: Draugh (1/3/SD), Bors (6), Oswin (7), Gilliam (8), Gatrie (9), Meg (10), Kellam (Awakening)
The Merric
Standard Issue Mage MK.5
Notable Merrics: Merric (SD), Lugh (6), Erk (7), Lute (8), Soren (9), Miriel/Ricken (Awakening)
The Minerva
Standard Issue Recruitable Enemy Wvyern Rider MK.6
Notable Minervas: Minerva (SD), Miledy/Zeiss (6), Heath/Vaida (7), Cormag (8), Jill (9/10), Haar (9)