Well, believe it or not I actually updated the Moogle page. There was a nice webpage that I saw once outlining the history of moogles. It would've been real nice for me to include it here, but now I can't find it. I guess I'll have to go from memory an personal experience. One not before I begin, all game numbers are in reference to the Japanese version since that appears to be the generally excepted number (even I have started to call FF:III (US) FF:VI).
So, from what I remember from the site, the beavers of Final Fantasy II can kind of be considered the precursers to moogles. I guess that's because they sort of look like the moogles in Final Fantasy III, but brown. As you can gather from that last sentence, moogles first appeared in Final Fantasy III. They haven't actually called them moogles in the game, at least not as far as I got so far (but that may be since it's translated from Japanese), but they obviously are moogles, as seen here. I also don't recall seeing moogles in Final Fantasy IV, which was somewhat disappointing, but I know for a fact that they are in Final Fantasy V, now part of Final Fantasy Anthology for Playstation. Apparently one of the character is friends with the moogles i nthis game. My personal favorite appearance of moogles is inFinal Fantasy VI (also part of Final Fantasy Anthology). Mog, in the above picture, is from that game. The background is also a representation of the moogles from that game, so thanks to whoever created it.
As the series moved to Playstation (grr...), the moogles' roles became diminished. In Final Fantasy VII, Caith Sith had his (supposedly) moogle robot and there was th magicite Choco/Mog. As for Final Fantasy VIII, the only moogle was the Minimog card, pictured above. Final Fantasy IX brought the moogles back, though. They played an important role by actually being the save points for your charcters. Beyond that, they delivered mail and ran the chocobo mini game, and Mog was a summon for Eiko. Unfortunately, IMHO, they don't look as good as they did in FF:VI. As for the most recently released Final Fantasy, Final Fantasy X, the moogle has a much diminished role again. It's only representation is as Lulu's stuffed moogle doll, which I call don't think looks that good...why put X's for eyes?
Of course, Final Fantasy is not the only Square seres to include the moogles. The moogles appeared in The Secret of Mana, but looked fairly different, brown with colored stomachs. And there is also Chocobo's Dungeon. I'm pretty sure that moogles are in the first one, but I'm certain that Mog is in Chocobo's Dungeon 2. Since Mog looks like he does in FF:VI, I'm also quite partial to his representation in this game.
Looking on the horizon, Final Fantasy XI, the first online Final Fantasy, is promising to have a Moogle house as your place to stay in your hometown. No pictures of them yet, but here's hoping that they look good. Also, the Square and Disney game that will be coming out, Kingdom Heart, definately has moogles. Unfortunately, they look a little the bodies seem small and the ball above them quite large, so it's not the best portrayal.