The game has so many Mythology Gags and Shout Outs, most of which were unfortunately lost in (mis)translation. To name but a few: (Warning: Unmarked spoilers below!)
Mount Gulug is a mistranslation of Gurgu Volcano from the original Final Fantasy.
The Eidolon Madeen is a mistranslation of Maduin, a reference to an Esper of the same name from Final Fantasy VI. Its attack was supposed to be Terraforming, but was mistranslated as Terra Homing.
The cry of the dwarves, "Rally-ho!", is a mistranslation of Lali-ho, the cry of the dwarves in Final Fantasy IV.
If you examine a sword that looks remarkably similar to the Buster Sword in the weapon shop in Lindblum, Zidane will remark "I remember a guy with spiky hair who carried something like this...", referencing Cloud from Final Fantasy VII.
One of Zidane's lines towards the end of the game is "No cloud, nor squall shall hinder us!"
A sidequest reveals that Princess Garnet's real name is Sarah, the same as the princess from the original Final Fantasy.
If you examine the grandfather clock in Quan's Dwelling, you are told that there is nothing inside. This is in reference to Final Fantasy VI, where there would be an Elixir in every grandfather clock in the game. This is the only place in Final Fantasy IX where there is a grandfather clock you can examine, so it was likely an intentional reference.
The original Final Fantasy also featured a villain named Garland.
When Princess Garnet is first encountered, she wears a robe like that worn by the White Mage in the original Final Fantasy.
The world map bears a striking resemblance to the world map in Final Fantasy I.
The Show Within a Show play "I Want to Be Your Canary" features a character named Princess Cornelia, a reference to Coneria (renamed "Cornelia" in later translations) from the original game. Since her father is named King Leo, this is also a reference to King Lear (see the entry on Shakespeare, below).
The four elemental fiends fought in Memoria originally appeared in the first Final Fantasy, though Marilith's name is mistranslated as "Maliris" in Final Fantasy IX (this fiend was called "Kary" in the NES version of Final Fantasy, though later translations used the name "Marilith").
Freya's can find a weapon named Kain's Lance, referencing Kain Highwind from Final Fantasy IV.
In one scene in the Black Mage village, a couple of Black Mages are talking about naming a chocobo "Bobby Corwen". The first syllables of those two words, "Boco", refer to Boko, Bartz's chocobo from Final Fantasy V.
The final boss, Necron, uses an attack called Grand Cross, the same name as an attack Exdeath uses in Final Fantasy V.
The Limit Break in this game is called "Trance", which first appeared in Final Fantasy VI as one of Terra's skills after she unlocks her Esper power. It was translated as "Morph" in the original SNES version of Final Fantasy VI.
Zidane's ultimate weapon is called Ultima Weapon, which is a recurring weapon in the series (it's also known as the Atma Weapon in the SNES translation of Final Fantasy VI).
Three of the attacks that Carbuncle can use, Ruby Light, Emerald Light and Diamond Light, refer to Ruby WEAPON, Emerald WEAPON and Diamond WEAPON from Final Fantasy VII.
Could be a coincidence, but one of the "SFX" moves during the stage fights at the beginning of the game (flashy moves that do no damage) is called "Meteo", which could refer to how the spell "Meteor" was shortened to "METEO" in the SNES version of Final Fantasy VI.
Cid's frog form bears a striking resemblance to Frog from Chrono Trigger.
Examining the fountain in Lindblum Castle gives the message "Looks like there's no place to put the medal here." Since there is no medal, this is a very clear Shout Out to a puzzle in Resident Evil 2.
Talking to a certain NPC in Lindblum will make him remark "Dammit, Jim, I'm a doctor, not a miracle worker!", referencing Bones McCoy's Catchphrase from Star Trek.
Necron quotes Yoda before the fight with it. ("Fear leads to anger. Anger leads to hate. Hate leads to suffering.")
After fighting Steiner for the first time, he collapses, remarking "Bah! Only a flesh wound!", a reference to the Black Knight scene from Monty Python and the Holy Grail.
There are a few references to William Shakespeare, namely that the prince of Burmecia is named Puck (the name of a character in A Midsummer Night's Dream) and the author of "I Want to Be Your Canary" is Lord Avon, a reference to Shakespeare himself who was often called the Bard of Avon.
Zidane having a monkey-like tail is a possible reference to Dragon Ball. There are strong similarities in backstory between Zidane and Goku.
Sir Fratley could be a reference to Michael Flatley, the Irish dancer renowned for the Riverdance.
Which is also referenced to the dancing Freya and the others do in one scene, which looks very Riverdance-esque.
Two workers in the Lindblum Synthesis Shop argue over whether or not it is better to have a strong weapon for power or a strong armor for safety. This is a reference to two officers from Parasite Eve (also made by Square) where they had a similar argument; one wanted stronger guns while the other wanted safer guns. Their names, Wayne and Torres, were reused for the characters in the synth shop.
"Gaia" and "Terra" are the Greek and Latin words for "Earth" respectively.
Each of the party members behaves like a specific job class from a previous Final Fantasy game. Zidane is a Thief with some Ninja-like abilities, Vivi is a Black Mage (obviously), Steiner is a Knight (with Magic Knight abilities if paired with Vivi), Garnet is mainly a Summoner with some elements of White Mage, Quina is a Blue Mage (possibly part Beastmaster, depending on how you view the similarity between hir 'Eat' ability and the 'Capture' ability of a Beastmaster), Amarant is a combination of a Monk and a Ninja, Freya is a Dragoon (dressed as a Red Mage!) and Eiko is mainly White Mage with some elements of Summoner.
There's also Beatrix, who is clearly a Paladin, albeit with much stronger White Magic than most of them tend to have.
Although not the most popular of the series, Hironobu Sakaguchi himself has gone on record to say that this iteration is arguably his favorite and represents everything he personally holds to be at heart of Final Fantasy.