Fantasy: magic is common and generally accepted
-Dark fantasy (not horror) close to horror but different,. Horror emphasis on fear while dark fantasy emphasis on despair (The Brothers Grimm, Van Helsing or the Witcher)
-Fairytale or light fantasy: not as tragic as the dark fantasy but no need to be funny (Shrek, the discworld)
-High fantasy/heroic : in another world with an epic character or story (Lord of the ring)
-Low fantasy : often in real world is generally humoristic or horrific (Conan the barbarian ?)
Science fiction: technology or social engineering is the central element
-Post (apocalyptic) : after the destruction of civilization (Fallout 3, 28 Days Later, the last of us)
-Cyberpunk: emphasis on artificial intelligence and cybernetics or transhumanism (Deus ex, Brave new world, Shadowrun)
-Hard (obey to a complex system and/or based on real technology knowledge ( Isaac Asimov)
-Soft (fringe science) an approximate conception of science
-Steampunk: emphasis on steam-powered devices with anachronisms and future technologies incorporated into the Victorian era (Final fantasy 9, Eberron, Jules Verne, H. G. Wells)
Science fantasy: mix of science fiction and fantasy (most of the Final fantasy games)
Fantastique: magic is uncommum and the unknown generate fear so it's similar to horror but not always( H.P. Lovecraft, Guy de Maupassant,Edgar Allan Poe)
Horror
Alternate reality or alternate history if it’s on Earth: no magic, same physic but with a different political/cultural background (Batman)
* Keep in mind that this list is non-exhaustive and of course it's also possible to mix the genres
*It‘s also possible to mix eras and have anachronisms just like most of the Final Fantasy games