The idea of the four elements earth, fire, air and water is from acient Greece, but while Greek mythology had nature spirits they weren't divided into these elements quite so neatly as FF's elementals. This idea seems to have been propounded by occultists later on, the earliest I know of being the Swiss alchemist Paracelsus, who also gave each type of elemental a name (gnomes for earth, salamanders for fire, sylphs for air and undines for water). Elemental spirits are said to take on any form, so none of the FF depictions can really be inaccurate.
Air Elemental -- Jinn -- Earth Elemental -- Fire Elemental -- Salamander -- Sea Elemental -- Water Elemental
Air elementals were referred to as sylphs by Paracelsus (possibly a combination of "silva", Latin for forest, and "nymphs", Greek nature deities) and are often also associated with fairies or elves (although the latter are more often considered earth elementals). They're considered to be knowledgable and gentle. According to the 19th century occultist Eliphas Levi, their king is named Paralda.
Earth elementals were referred to as gnomes by Paracelsus (a word which is now used for just about any kind of dwarfish creature from folklore. The word is derived either from "gnomus", which I believe is New Latin for "dwarf", or from "genomus" Greek for "earth dweller") and are often also assosiated with satyrs, elves, leprechauns and dryads. They're considered to be trustworthy and mostly benevolent. According to Eliphas Levi, their king is named Gob.
Fire elementals were referred to as salamanders by Paracelsus (unlike the names he gave to the other three elementals, this one wasn't coined by him, it's a reference to folklore - see the salamander entry) and are often associated with wills o' the wisp. They're considered to be short-tempered and energetic. According to Eliphas Levi, their king is named Djin (again, taken from mythology).
IMAGES: Illustration of a salamander from Paracelsus' Auslegung von 30 magischen Figuren -- Illustration of a salamander from a 13th century bestiary (MS. Bodley 764) AIR ELEMENTAL

EARTH ELEMENTAL

FIRE ELEMENTAL
