Atlantis is a city that is believed to have been lost long ago where all the good sea creatures stay in the afterlife, which makes it the underwater equivalent to Heaven and the opposite of Davy Jones' locker. However, it remains intact and is currently ruled by Lord Royal Highness. It first appears in the episode "Neptune's Spatula."
The Atlantis in "Neptune's Spatula" looks different from the city in the episode "Atlantis SquarePantis." They either have drastically advanced in technology in the space between the two episodes or they are two different places altogether. Atlantis, as seen in "Atlantis SquarePantis," is the mythical Atlantis itself and has no relation to Bikini Bottom whatsoever. The royal government was totalitarian.
The inhabitants in "Atlantis SquarePantis" have put away all weapons in an immense vault and maintain a monarchy, seeing how the Lord Royal Highness is their kind. The Atlanteans traveled countless light-years from their home planet and brought the oldest living bubble on the journey, living over a million years. The Atlanteans took great advances in weapons, wealth, science, art, and things that people took for granted in Bikini Bottom today.
According to the real Atlantis legend, the buildings were made of stone. None of this was seen, for the artists decided to put futuristic designs and shuttles in this episode, unlike how the animators of Atlantis: The Lost Empire put their civilizations and designs where they had excluded designs for ancient Greece and Rome.