Born as a visual novel created by TYPE-MOON and Takeuchi Takashi, the Fate series is one of the most popular franchises in Japan with many different anime, manga, novel, and game adaptations.
The main series of Fate is the popular Fate/Stay Night which also includes the third route of the story called Heaven's Feel which was later adapted in three movies by the animation studio UFOTABLE.
Fate Apocrypha is a spin-off of the main series that tells a story in a parallel world to Fate/Stay Night that gained a lot of popularity thanks to ANIPLEX anime adaptation made by studio A-1.
Fate EXTRA and EXTELLA are part of the same world which is completely different from that of Fate/Stay Night and was mainly adapted with various videogames and a recent anime adaptation by studio SHAFT.
Fate/kaleid liner PRISMA☆ILLYA is a completely different story with some of the same characters of Fate/Stay Night in a Magical Girl setting.
Fate/Grand Order is a mobile game and currently the most popular series of Fate with many of the latest figures coming from it and a very large pool of new characters designed by popular illustrators that get constantly added to the game.