Alexandria is the largest kingdom on the Mist Continent in Final Fantasy IX. It is the home of several of the major characters in the game, including Garnet, Vivi, and Steiner, among others.
Alexandria has the most cities of any of the Mist Continent kingdoms, with Dali, Treno, and the eponymous Alexandria. There are various other landmarks, such as the Ice Cavern, Evil Forest, a Qu's Marsh, and Quan's Dwelling. Most of the first disc of FFIX is played in Alexandria, where it is a central part of the storyline.
The games starts with the party in Alexandria, as a botched kidnapping attempt results in a crash landing the in the Evil Forest below the major city. From here it's off to the Ice Cavern, then Dali, and finally to the neighboring kingdom of Lindblum. Garnet takes Steiner and returns to Alexandria, intent on stopping her mother's warmongering. Zidane and the group arrive in Alexandria after teleporting from the Red Rose, and wind up saving Garnet. Several party members remain in Alexandria to fight off their enemies, and return to the party following Garnet's return to become the new Queen.
As Garnet prepares to become the new Queen of Alexandria, it comes under attack from Bahamut. Eiko and Garnet's magical jewels combine, the castle transforms, and Alexander (possible explanation for the city name) is summoned. Much of the city is destroyed, as are both of the Eidolons.
After this, most of the game takes place outside the kingdom.