I'll limit my list to include only sentient species.
Here are the ones from the games I've played...

FFIV: Lunarians, Summons.

FFV: Moogles, werewolves.

FFVI: Espers, werewolves, sasquatch, imps.

FFVII: Cetra, AKA "the Ancients"; Red XIII's tribe; "Toysaurus" (apparently intelligent robot animals).

FFVIII: Moombas (as gleefully mentioned by Mr Pies), Shumi, Guardian Forces.

FFIX: Genomes, Burmecians/Cleyrans, Black Mages, fish people, hippo people, green-furred conical people, moogles, demi-human rodentoid people (e.g. Baku, Dr Tot).

FFX: Hypello, Guado, Ronso. (Al Bhed are simply a sub-group of humans, so I don't count them as a different race).

FFX-2: Same as above, with the addition of Cactuars and, more significantly, Cactuar Mothers.

Certain I've missed a few, but I'm sure those gaps'll get filled by others. I'll add later any that I know I've missed.