Baphomet is one of the Demon Princes of the Tanar'ri in Dungeons & Dragons. Hailing all the way back to 1st edition, he is one of the original Demon Princes created by none other than Gary Gygax, and specifically owes his name to real-world demonology. Baphomet is the patron god of the Minotaurs in most settings, although with his title, he could certainly be given authority over all kinds of beast-man species. He has a long-standing rivalry with Yeenoghu, though exactly why has never really been made clear (and has actually been forgotten by both as of 3.5, according to Hordes of the Abyss). Oh, and he also hates Graz'zt, but then again so do most all of the princes.
In D&D specifically, Baphomet owes his inspiration to the stories of the Templars, who were said to worship a demon named Baphomet via the icon of a golden bull's head (Either a corruption of the word Mahomet, or Arabic for Abu Ifamat, Father of Understanding). Consequently, the D&D Baphomet is depicted as a massive and powerful minotaur, furthering his connections to that race. 5E uses a combo of his classic minotaur look with the flame on his head based on the demon he was based on. He rules Abyssal layer #600, The Endless Maze, which is interestingly also the ancient home of the obyrith Pale Night, with whom he has some kind of arrangement so that the leave each other alone.
In Pathfinder, due to legal issues, Baphomet is instead styled after the more general depictions of his namesake, such as the titular character on the Devil card in the Tarot's Major Arcana; he appears as a lean, humanoid goat with a brazier-like burning third horn jutting out of his head and an upside-down pentagram on his forehead. Amusingly, this is worked into his lore; long and short of it, he used to be the first minotaur and one of Lamashtu's lovers/sons, until he tried to impress her by screwing over Asmodeus. The king of devils enslaved Baphomet, branded his mark into Bapho's skull and then tossed him into an extradimensional maze to die. Joke was on him, though; although he didn't do it until after he'd mutated into his present state, Baphomet eventually came to understand the maze so well he was able to not only leave, but to steal it away from Asmodeus and merge it into the Abyss, founding his own fiendish kingdom.
This emphasizes a trait that both versions of Baphomet uphold: despite his title of the Demon Prince of Beasts, Baphomet is no idiot. In fact, he's actually really smart, and enjoys labyrinthine, almost diabolic plots and schemes, although always with the ultimate goal of tearing down and subverting civilization so that his beast-man servitors can rule over the rubble. This is an important distinction from his rival Yeenoghu, who is rather stupid.
As proof of his intellect, 3.5 indicates that part of his layer of the Abyss houses the Tower of Science, where he literally spends most of his time perfecting horrific experiments to create all new creatures (usually demons, though the big guy isn't picky if the creature proves effective). Witness the Baphitaurs, the result of his efforts in adding fiendish blood to minotaurs to create minotaur-tieflings.
|The Demon Princes of Dungeons & Dragons|
|Tanar'ri:|| Baphomet - Demogorgon - Doresain - Fraz-Urb'luu - Graz'zt |
Juiblex - Kostchtchie - Lolth - Malcanthet - Miska the Wolf-Spider
Orcus - Sess'innek - Shaktari - Turaglas - Yeenoghu - Zuggtmoy
|Obyrith:||Dagon - Obox-ob - Pale Night - Pazuzu - The Queen of Chaos - Sertrous|