Adventure Path
Adventure Paths are adventure modules linked together by an interconnected story arc, with each adventure moving the players through a pre-written campaign from beginning to end. The term was popularized by Pathfinder and Paizo's earlier Dungeon Magazine adventures, but predates it; perhaps one of the oldest examples of this was the Dungeons & Dragons AP "Against the Giants", a linked series of modules that led to the players fighting multiple species of giants before ultimately battling against the drow who had roused them to war.
Dungeon Magazine List
The Shackled City
Someone thought an inactive volcano was a good place to make a town. The PCs start of by looking for some kidnapped children and stumble into far greater conspiracies. Mostly urban based around a single city, with the outlying areas mostly being fluff and where you can make shopping trips to. Was latter compiled into a book with some fixes and new content.
The Age of Worms
In the shithole mining town of Diamond Lake, some newly unearthed archeological finds threaten to unleash some long sealed horrors upon the world once more. Centered around a "Free City" which is really the Free City of Greyhawk with the serial numbers filed off, but can be replaced by any campaign specific large city with Waterdeep and Sharn given as examples.
Since the majority of the campaign is urban, and travel is either off screen or teleportation based, it's less setting dependent than other adventure paths. Indeed, unlike most published adventures that aren't explicitly Eberron based, it works surprisingly well in the setting. The official conversion gives instructions for an interesting twists on generic ally NPCs (and explains where they came from). The name of the disaster the players are trying to avert also plays well with Eberron's epoch naming.
Savage Tide
The Pathfinder List
Rise of the Runelords
Pathfinder's 'flagship' AP, RotR hits a lot of more classic tropes. Starts off with the party fighting goblins and an evil Aasimar, moves on to fighting ghouls and a Lamia, then it becomes a mild wilderness game to fight your way to a big ass dam and fight several types of ogres and giants, save a few towns, then go off to fight dragons and ancient evil spellcaster. Initially released for 3.5 before getting an official second edition for Pathfinder.
Curse of the Crimson Throne
An Evil Queen and intrigue in and around her court. Popular enough to get an update from 3.5 to Pathfinder just like RotRL did. Also fucking impossible. Do not attempt if your GM has any sort of malicious intent. Fucking bards, man.
Second Darkness
Drow want to blow everything up, and the elves who send you to stop them are kinda jerks. Not one of the most popular APs.
Legacy of Fire
Genies and Arabian Nights the adventure Path. The final 3.5 AP
Council of Thieves
The first AP written for the Pathfinder rules instead of 3.5. Be a hero and help the Nazis against the Mafia. Huge city but a total shithole. Has some major issues with player involvement because a lot of shit happens behind the scenes and the AP doesn't think much of informing the players about it. Needs some work from the DM. Starts off like a rebellion story, but does a hard turn into something else the last two books so if you wanna do that, play Hell's Rebels instead.
Kingmaker
Make Your Own Kingdom: The Adventure Path. Introduced the idea of an AP specific minigame. Unlike the future ones this one actually works well enough to bother with it (it was updated in Ultimate Campaign to plug a few issues). The main complaint outside of the kingdom rules being imperfect is lack of foreshadowing for the true big bad: For the first few books the only clue you'll find is a non-decaying, carrion repelling unicorn corpse in the middle of the wilderness (Not an exaggeration).
Probably best known because a Owlcat Games (a Russian game developer), with Chris Avellone leading the narrative design, got the rights to make a CRPG based on this and released it on October 2018.
Serpent's Skull
The jungle exploration path with lizardfolk, normally considered one of the weakest APs alongside Second Darkness and Wrath of the Righteous due to the tremendous amount of padding.
Carrion Crown
The classic horror monster path
Jade Regent
Travel to Tian-Xia to overthrow the current regime and install its' proper inheritor who happens to be one of your bffs. A lot of the path is just getting there. Includes rules for managing a caravan, but they're so totally pointless you can just ignore them and the campaign is completely unchanged. Also includes a romance system for NPCs, with the particular odd bit that it is easier to get in a relative's pants because your relationship starts closer and nothing actually states the obvious to stop it.
Skull & Shackles
Pirates! You start pressganged on a ship, and after breaking free you need to make a name for yourself on the high seas. Awful for a GM that doesn't want to spend hours voicing the dozen or so important NPCs appearing in just book 1. Uses naval combat rules, though ship HP is so inflated boarding is the only practical option without house rules.
Shattered Star
Semi-sequel to Runelords, you need to assemble a powerful artifact and a different villain of ancient Thassilon threatens to rise. Almost pure dungeon crawl.
Reign of Winter
Somebody turned down the thermostat in Taldor, so you go and investigate. Things get out of whack when you fight winter fey in the middle of summer, get teleported to Bumfuck, Icy Nowhere and get roped into a date with a Winter Wolf, and it only escalates from there when you go to another continent, another planet, and another galaxy on your quest to rescue damsel-in-distress Baba Yaga from her uppity daughter and son, Rasputin, and his army of tear gas elementals, WWI Russian soldier zombies, and magically-animated tanks. If your group hates fun, the author for the book with that last bit wrote an unofficial conversion that replaced going to WWI Russia with a trip to Greyhawk to fight Iuz.
Wrath of the Righteous
The Demons of the Worldwound have decided that being trapped in the area around the Worldwound is a bum game, and break free, inadvertently giving your party mythic power in the process. You progress deeper into the Worldwound and eventually to the Abyss itself to save the world from demonic invasion. Dependent upon poorly balanced mythic rules (essentially a redo of 3E's Epic Adventures) and notorious for some very mismanaged NPCs. Unlike most APs, this one goes all the way to level 20.
Mummy's Mask
Ancient Egyptian/Osirion path with ancient tombs, mummies, and floating pyramids.
Iron Gods
Spiritual successor to Expedition to the Barrier Peaks. Travel through Barbarians and scavenged technology land fighting aliens, an organization that wants to monopolize all high technology, and face down an AI that wants to make itself a god. Do not charge the final boss if it's linnorm is still alive. Learn from others mistakes.
Also a steampunk twist on this campaign would be amazing.
Giantslayer
Big Stuff. Kill it fast before it kills you. Do no attempt to make giants fail will saves. The GM will fuck your ass.
Also, the 5th book is just a fire-themed version of the 4th.
Hell's Rebels
In a city at the edge of Cheliax, the Thrunes moved one of their family that even those devil-worshipers find too much. As he cracks down, you rise up. Well developed city. This AP actually is what most people thought Council of Thieves was going to be.
Hell's Vengeance
The bad guy path - Cheliax can't focus on the Hell's Rebels rebellion because they're dealing with one of their own, an ill-considered attempt at their capital. You work for the Queen in helping put it down and restoring order in the lands. Pretty damn great if you have an average to good GM.
Strange Aeons
Lovecraft the adventure path. Lethal as fuck. Bring backup characters (which is weird, because the plot is heavily dependent upon the original PCs) and do not, I repeat DO NOT have a mental score below 10. You'll be insane by the end of the first book. Also bring a tank along with you everywhere.
Ironfang Invasion
One of the more "traditional" recent adventure paths, this one involves stopping a massive hobgoblin army, though it does have some relatively unusual twists in the later books. Set in the country of Robin Hood (or Wood, for those of us who live in T.H. White's books).
Ruins of Azlant
Visit the ruins of not-Atlantis and fight evil psychic fish (Aboleths), lots of nautical themes and aquatic monsters. Also, there's a magic theremin as an optional treasure.
War for the Crown
There's a Succession Crisis in Taldor, so it's time to play the Game of Thrones... or it would be if it could decide what the hell the main conflict is supposed to be. Women can't inherit, except half the characters are women who inherited their position. A dozen factions are involved in the power struggle, but you are literally only given more details than the names of three of them, and one of those three doesn't show up till almost the end.
Return of the Runelords
A sequel to Raise of the Runelords and Shattered Star. You know those ancient max level wizards who ruled the world millennia ago? They're back! Goes to 20, which it does by excluding a lot of the non-adventure parts of each book that were previously included.
Non-Paizo Adventure Paths
Dawn of Defiance
A free series by Wizards of the Coat for Star Wars: The Roleplaying Game Saga Edition. Shortly after Revenge of the Sith Bail Organa gathers allies to make the first strikes against the new Empire, starting with dealing with the Empire's mysterious "Sarlacc project". Since at the time (and even afterwards) the intertrilogy era had little going on in it the writers were allowed to go nuts and make a love letter to the Star Wars Expanded Universe.
Scales of War
After taking over Dungeon Magazine from Paizo and continuing it as a digital only publication, Wizards of the Coast tried their hand at an AP of their own called Scales of War. It's largely forgotten because it was published in the digital only Dungeon issues and for 4th Edition Dungeons and Dragons. Even those that did like 4E weren't too fond of it, calling it disconnected, repetitive, and pure dungeon crawl.
Way of the Wicked
An adventure path devoted to playing evil characters. Notable not just for being made for evil characters, but portraying various evil schemes from the other point of view. Break out of jail! Sabotage a town from the inside! Run an evil lair and conduct evil rituals! Slaughter towns! Rule over your conquered kingdom! A shame the art isn't that great.
Predates Hell's Vengence and indeed likely part of why it was that exists. This series repeatedly topped the best selling third party Pathfinder products.
The same creator made two books into a second all underground adventure, but it since wound up vaporware.
The Northlands Saga
Relatively low magic viking adventure with a strong emphasis on Norse culture. Latter books are weaker than the earlier ones because they were made from an outline and notes. Includes time skips of significant length (several years). Since it's light on magic items and high on money, bring a crafter wizard.