Some translations of Final Fantasy IV have Cid ask you about his airships, guessing that Cecil and his "goons" must have "wrecked them up something awful".
Biggs and Wedge: Biggs and Wedge are intentional shout outs to Star Wars, as they are Luke's wingmen in the first released movie. Their original appearance in North America had the names mistranslated as Vicks and Wedge in the SNES versions of Final Fantasy VI and Chrono Trigger, but was corrected in their remakes, as well as in other games where the characters appear, such as Final Fantasy VII, VIII, and X.
Bartz's mom is named Stella. So, when she collapses in his flashback, what does his father yell? That's right — "STELLLLLLLLA!!!"
In the Advance version, one of the new jobs is Gladiator. The description of the Gladiator's !Finisher ability says:
His first Blitz, Pummel/Bakuretsuken, is Ryo Sakazaki's Zanretsuken, with the same controller motion.
In Final Fantasy VII, a boy in the slums pretends to be a train operator, obliviously not realizing that he's in Midgar, and it's not as clean as he'd hoped; this is an obscure reference to Akira Kurosawa's film Dodesukaden.
Final Fantasy X: Shelinda tells Tidus that Seymour is headed for Macalania Temple. When Tidus informs the others, he messes up the temple's name, resulting in this bizarre (and dated) conversation.
Tidus: They say Seymour's headed for Macarena Temple. Wakka:Macalania Temple. Tidus: AYE!
Even more bizarre is the fact that that exchange happened in a game released in 2001, whereas the Macarena was popular in 1995 and 1996. That's right: the Shout Out was already dated the second it was made.
A recent update in Final Fantasy XI included a reference to Kyubey of all things, as the Tarutaru for the limit break quest to get past level 90 says "Are you ready to sign a Contractaru with me and become a magical g—er, a mightier, more majestic adventurer?" and after you finish the quest, you get the key item "soul gem".
Final Fantasy XII really enjoys doing this. The most obvious ones come in the Gilgamesh fights. As you battle him, he pulls the swords of previous Final Fantasy characters out to attack you with, including the Buster Sword (which has the kanji for "Replica" spray-painted on its side), Brotherhood, and Odin's Zantetsuken. For a change of pace, he also breaks out Loto's Sword.
Just in case it wasn't clear from the Buster Sword, every one of these swords is a fake and has a pretty obvious tell that gives it away. The Revolver has the wrong symbol and no trigger, The Orichalcum is too long, etc.
Final Fantasy Tactics features two characters named Wiegraf (Wiglaf) and Beowulf, obvious references to the poem Beowulf. The two FF characters have no connection however, other than the fact that Beowulf (the FF character) is a Palette Swap of Wiegraf.
The Redwings from Final Fantasy Tactics Advance is the shout out for Final Fantasy IV's Red Wings. The Redwings' leader Grissom is a Dark Knight, but his sprite and attacks are based on a Paladin. This is a shout out to the Red Wings' leader Cecil Harvey.
The game also has the "kings", five wizards with (horribly Anglicized) French names for the five colors of magic: Ruuj (red), Bliu (blue), Verre (green), Nware (black), and Blanch (white). While we're here, those are also the three colors in Magic: The Gathering, though they have different meanings there. (For starters, red magic in M:TG is what black magic is in Final Fantasy.)