The name Final Fantasty
Some recuring themes and characters, (CId in every game and Gilglamesh, summons Bahamut)
They are all made by the same developer, Square-Enix (Previously SquareSoft).
As regards to storyline, they share many plot devices (crystals, summons, magic, etc.), but are not linked in any way.
Originally "Final Fantasy" was to be the last game Square produced, but because it was extremely popular, it kept the company from going under. Thus all Final Fantasy games numbered after "I" are, in some ways, a sort of joke, as if "XIII" were the final swan-song of the company...but really not.
Exdeath: The VOOOOOOOOOIIIIIDDDD!
There are returning features like black and white magic, crystals, chocobos, reccuring enemies and so on, but they are also connected with the interdimensional rift. Gilgamesh travels many FF - worlds via the rift.
Duodecim does a pretty good job of explaining it.