Pathfinder is often conflated with D&D, 4e I think, but it isn't actually named "Dungeons & Dragons." What is Pathfinder's relationship to D&D, and how does it fit in with the various D&D editions?
Is Pathfinder D&D? No, but kind of.
Pathfinder is published by Paizo, who does not own the rights to Dungeons & Dragons. Those rights are owned by Wizards of the Coast, who currently publish D&D 5e. But Pathfinder is a spin-off of Dungeons & Dragons, specifically D&D 3.5e, and is extremely similar to that game in many ways. Playing a game of “3.PF,” using material from both rulesets, is quite possible and popular.
How and why this came to be, however, requires a history lesson.
D&D 3.5e and the Open Game License
Wizards of the Coast released the foundation of the D&D 3.5e ruleset under the Open Game License, which was very, well, open about how much of it could be re-used. This led to a huge explosion of third-party content for 3.5e, and 3.5e lived a rather-long life as these things go. There was a ton of material for it, the people playing it had gotten very used to dealing with, or even getting attached to, its myriad problems.
At this time, Paizo published the Dragon and Dungeon magazines under license from Wizards of the Coast. They also published a fair amount of their own adventures for the 3.5e ruleset, under the Open Game License.
D&D 4e and the non-open Game System License
Then Wizards of the Coast released D&D 4e. The fourth edition of the game was a massive departure from previous editions of D&D, and was extremely controversial. Many players had no desire to switch to 4e, and continued playing 3.5e. Some even decided they didn’t like Wizards of the Coast’s D&D altogether, and went back to older editions of D&D. And many did play 4e, and there are some hints that 4e did relatively well in bringing new players into the game.
So D&D had fractured its fanbase, and there were a lot of people playing D&D but not playing the edition of D&D that Wizards of the Coast was actually publishing.
At the same time, Wizards of the Coast got a lot more possessive with its property. They did not renew their Dragon and Dungeon licenses with Paizo, again publishing those in-house, and they did not release 4e under the OGL. Instead, they created and used the Game System License, which is vastly more restrictive than the OGL was. This made it nearly impossible to develop 3rd-party content for D&D 4e.
This put Paizo into a very tight spot: with their magazine revenue taken away, the latest edition of D&D hostile to third-party content, and the edition of D&D that was their bread and butter, 3.5e, aging and slowly dying, they had a serious problem.
Pathfinder, built on 3.5e open game content
Pathfinder was Paizo’s answer. It was based on the open game content from D&D 3.5e, and pushed hard to capture the market of people who refused to play 4e and were sticking with 3.5e. By promising 3.5e-but-better, and by delivering fresh content, Paizo could keep 3.5e alive, and therefore continue to make adventure material and maintain that revenue stream.
This worked. Through a phenomenal hype machine, Paizo could offer a game system that amounted to a few houserules applied to 3.5e, call it “better,” and capture a pretty large market share. It cost them relatively little to do it, and it allowed them to continue to publish their adventure modules, which were their real focus and interest. Extensions to the Pathfinder system (classes, feats, and so on) were enabled through low-paid freelancers with minimum editorial oversight, and allowed Paizo to keep Pathfinder “alive” through a blistering release schedule, again at relatively low cost. And so they could focus on selling adventures.
An aside: the “old-school renaissance”
Paizo wasn’t the only company to notice the fractured D&D fanbase. Numerous other games, labeled “OSR,” came out to try to capture those players who ditched not only 4e, but 3.5e as well. So in addition to Pathfinder being a spin-off of 3.5e, there are other games that are spin-offs of or inspired by older editions, from before Wizards of the Coast bought the rights to D&D.
The death of D&D 4e
The story of Pathfinder suggests that D&D 4e was a complete disaster; that’s certainly how many Pathfinder fans view it, and probably also Paizo themselves. However, it’s not really accurate: D&D 4e did well enough, and again did particularly well with new players, relative to Pathfinder mostly focusing on old players who didn’t like 4e.
As a game designer, I will also say that D&D 4e is easily one of the most tightly-designed RPGs in existence. Other editions of D&D aren’t even playing in the same league. That’s not everything, not by a long shot, but a lot of the criticisms leveled against it were based on perception from quick reads of the book, and not from actual play.
But there were also a number of large problems. Some of it was poor planning, some of it was pressure from Hasbro (who owns Wizards of the Coast) to cut costs on D&D, at least part of it was a murder-suicide (!) by one of the lead developers of a 4e virtual tabletop, killing not only himself and his wife, but also that project and a lot of Wizards’ plans for 4e.
In the end, 4e ended up losing support from Wizards of the Coast, and even if you liked it sooner or later the fact that there was new Pathfinder content and no new D&D content meant a lot more people switched to Pathfinder.
D&D 5e: A return to form
D&D 5e was an attempt to recapture the player base that had been lost to Pathfinder and the OSR. It undid a whole lot of changes made by D&D 4e. It embraced an “old-school” playstyle to a large extent. It also put a huge emphasis on being simple, easy to learn and play, and welcoming to new players, which is not something any of the other games mentioned here can say.
And it has been extremely successful.
No numbers are known here, but recent years have been some of D&D’s best—and that goes all the way back to the original editions in the 70s and 80s that became an international phenomenon.
Pathfinder 2e: A surprising departure
Paizo is currently testing their second edition of Pathfinder. It is surprisingly 4e-like in a number of ways, which is somewhat ironic considering that Pathfinder was written as a response to 4e in the first place. (It also has a number of 5e-like features, and of course a whole lot of it is unique.) Perhaps most notably, it’s a large departure from PF 1e, greatly changing the game in a large number of ways.
This has been controversial.
They’re still in playtesting (and uncharacteristically open to feedback, from what I can tell), but They just officially released it, and there is a risk here for Paizo that they will follow in 4e’s footsteps—clearly not their goal. Time will have to tell on that, though—it’s much too soon to write the book on PF2e.
Technically: No, Pathfinder is not D&D.
Colloquially: Yes, many would consider Pathfinder a form of D&D, as Pathfinder (1e) is a direct descendant of D&D 3.5, so interconnected that many refer to it as "3.75" or "3.P". Be aware that purists on both sides may disagree with this answer, as it is a bit of a contentious issue, and a lot of people feel a need to point out the technical perspective that no, PF is not D&D.
The reason many would consider it D&D and label it as "3.75" or similar is fairly simple. Upon the creation of 4e, many considered the diverging nature of the game to be very far from what "felt" like "D&D", and at the same time, Wizards of the Coast dropped a lot of work that they had done with Paizo (such as the at the time age-old Dragon Magazine), "tightening up" their own hold on the IP and the constituent parts of the overall franchise.
This lead Paizo, spurred on by a lot of people disappointed in the development of 4e, to create Pathfinder, which many fans came to consider "more D&D than D&D".
But again, no, not technically D&D. Different company, different name, different owners.
Pathfinder is a continuation or offshoot of the 2003 D&D 3.5 ruleset, by the former publishers of Dragon magazine.
Wizards of the Coast (WotC) released Dungeons & Dragons third edition (3e) in 2000, having bought out D&D's original publishers, TSR, in 1997. Wizards continued to publish TSR's official Dragon magazine and Dungeon magazine until September 2002, when they leased the magazine rights to a third party named Paizo. Rumour at the time was that WotC's corporate bosses insisted magazines were no longer profitable, something Paizo would go on to disprove.
In March 2003, Paizo's Dungeon magazine (Issue #97) published the first module in the Shackled City Adventure Path, a new concept where the magazine released a continuing series of adventure modules, one per issue, which took the characters from levels 1 to 20.
Shortly after this, in July 2003, WotC released a revised D&D third edition (3.5). Paizo would go on to release two more Adventure Paths: Age of Worms, and Savage Tide.
Pathfinder Adventure modules
In April 2007, WotC announced that it discontinuing their lease on the magazines. The public officially learned why in August 2007, with WotC's announcement of D&D 4th edition. The final print issue of both Dragon magazine and Dungeon magazine were in September 2007.
Many people still wanted to play D&D 3.5 adventures, but the terms of Paizo's lease forbade them from simply continuing to publish their own magazines under another name. However, it did not forbid them from publishing a continuing series of adventure modules compatible with D&D 3.5. Paizo also held significant goodwill with former Dungeon magazine freelance writers and artists.
The result was the Pathfinder adventure path series, beginning in August 2007 with Rise of the Runelords Adventure Path. Initially, this series used the D&D 3.5 ruleset, which remained popular even after WotC's release of D&D 4th edition in June 2008.
Pathfinder Roleplaying Game
However, now that D&D 3.5 was out of print, Paizo decided to publish their own variant of D&D 3.5, known as the Pathfinder Roleplaying Game or Pathfinder RPG. This was possible thanks to the Open Game License, which allows third parties to republish much of the D&D 3.5 rules.
This ruleset, Pathfinder RPG, is now usually what people are talking about when they say Pathfinder. It was released in August 2009, along with the first part of the fifth Pathfinder Adventure Path, Council of Thieves, which was the first to use the Pathfinder RPG instead of D&D 3.5.
D&D 3.5 and Pathfinder are overall quite similar, except that Pathfinder has numerous changes or improvements. It has been nicknamed "D&D 3.75" as a result. It's sufficiently compatible that you can use D&D 3.5 material with Pathfinder RPG with little or no conversion. Many fans see it as the rightful successor to D&D 3.5, and at some points prior to the release of D&D 5th edition in 2014, Pathfinder actually outsold D&D.
Paizo currently plans to release a second edition of Pathfinder RPG in 2019.
Pathfinder is D&D with the serial numbers filed off
Pathfinder is not D&D in the same way a generic tissue is not a Kleenex, a copy machine isn't a Xerox, and a self adhesive bandage isn't a Band-Aid...
Which is to say, it is effectively the same game in most important regards, but it's not the "name brand version". It has it's own world, it's own pantheon of gods, and many other divergences from D&D 3.5. It is often referred to as D&D 3.75 because it improved on the previous edition of D&D at around the same time that 4th edition came out.
All in all it's much closer to D&D than say Dungeon World or Tunnels & Trolls.