Shore leave fics. There are tons and tons of stories of sailors Horatio and Archie going on shore leave and spending some quality time together. Subtypes include drinking at inns, sharing bed at inns, flirting with wenches, seducing lasses, ignoring gals (and lads), celebrating promotions, celebrating holidays together (often Christmas), road trips in carriages, visiting either family together, going to theatre to see a play (bonus if it's Drury Lane in London, double bonus for seeing one of Shakespeare's plays, triple bonus if they see or meet Kitty Cobham, and you have a bingo if they see Kitty Cobham actually act in Shakespeare's play in Drury Lane). Sometimes they split up and one of them stays at the inn in the port and the other has an adventure of his own and they reunite later, but most often they stay together.
Theatre fics. These stories deal with Archie's days spent at theatre. In one episode, he says he knew Drury Lane like it was his home. Theatre Royal, Drury Lane is a theatre in Covent Garden, London. Archie is often a lord's son with a rebellious streak who hangs out with actors or becomes a teenage actor. Once his family has had enough of his antics, they ship him off to the Navy.
Shakespeare fics. A simple set-up: Archie Kennedy quotes/loves/appreciates William Shakespeare. It may appear just as a dab about his love of theatre, but often the whole fic revolves around the Bard and it's full of allusions and quotes.
Jack Simpson/Archie Kennedy fics. What exactly has Jack Simpson been doing to Archie before Horatio came aboard HMS Justinian? Usually it's explicit sexual abuse. Simpson might be shown to have some genuine, yet extremely twisted affection for poor Archie but he's almost always monstrously sadistic and violent.
'Simpson's dead' fic. Hurt/Comfort Fic when Horatio tells Archie that Simpson is dead (inexplicably never shown onscreen).
Spanish prison fics. Stories about Archie and Horatio's reuniting in Spain. Most focus is on the interesting times after their voluntary return because Horatio gave Don Massaredo his parole. They share a cell as before, but now sans surly Hunter!
Comforting Midshipman Wellard fics. Hurt/Comfort Fics of Archie helping young Midshipman Wellard after he is unfairly beaten because of their lunatic of a captain. Horatio or Mr Bush might assist him as well, but the focus is usually Archie because it mirrors his situation in the first series when he suffered the most.
Briliant doctor fics. Fix Fic about a competent surgeon who saves Archie's life. Quite often it's the Age of Sail Friendly Fandoms crossover, and the briliant doctor is Stephen Maturin from the Aubrey-Maturin series. But the doctor can be also an Original Character or even dr. Clive from canon.
'Archie leaves the Navy' fics. Archie leaves the Navy before their transfer to the Renown and their adventures in the Carribean that result in his death in canon. Some of the stories are Fix Fics, but a number of them had been written (or at least started) before the episodes were released.
'Archie and Horatio live together as lovers' fics. Archie and Horatio leave the Royal Navy for some reason (injury, illness, disillusionment, hiding from the justice, "just because they want to") and live together as lovers in a lovely cottage somewhere in England (a house in London is common, too). Sometimes they both had a fulfilling career in the Navy and resign as older gentlemen. They have been together almost all their life (on ships and otherwise) and they settle together. It's usually full of sweet domestic bliss (very Jane Austenesque).
'Lord Edringon & the Kennedys are friends' fics. Word of God has it that Archie Kennedy is "the third son of a Scottish lord". Fanfics often reflect this. He's almost always from the upper class and put in contrast with Horatio (who is a doctor's son and a commoner). Lordship and other top-tier aristocracy is common, but Archie might also come from the lesser nobility (his father can be a baronet or just a land-owning gentleman with no title). It often turns out that Kennedy family mansion is not that far from the Earl of Edrington's estate (i.e. Major "My Lord" Edrington from "The Frogs and the Lobsters"). Friendly acquaintance and intercourse (social! social intercourse) between those two families ensue. Fics usually introduce several female original characters — sisters, wives or love interests for the guys.
Mythological creature fics. Horatio (or less frequently Archie) is seduced and/or possessed by a mysterious mythological being, usually connected with sea or water (a selkie, a rusalka and similar creatures).