Fairies?

Did we ever learn exactly what these creatures are?

They're arcanima constructs (living aether fashioned into an entity), not all that dissimilar from Carbuncles. How they're summoned is what's different. Carbuncles are manifested using complex mathematical formulas that determine their appearance and behavior. Every Emerald Carbuncle seems to be summoned fresh each time, and two will behave the same way; it's all pre-determined by the mathematician that designed the arcane geometry.

Faeries, however, are created once and then bound to a soul crystal. When the faerie is conjured forth, her experience becomes part of the memory of the soul crystal just like the scholar's. Because of this, every faerie develops a unique identity, and when the crystal is passed from one scholar to another, she goes with it. That identity, however, can be expressed in multiple ways; the fairy that you use can currently manifest as Eos (sunlight) or Selene (moonlight), but either way she's still Lily, the faerie from Surito Carito's soul crystal.

If you were going to try to biologically classify one using the Raimdelle's system (the most common system of taxonomy in Eorzea), they'd be categorized with sprites and such under as Elementals, a class in the Transcendent kingdom.

So why did the Nymian scholars model them after the Elementals?