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.
The first time it was shown was in the episode "Neptune's Spatula." The Atlantis here looked different from the city in "Atlantis SquarePantis." They either have drastically advanced in technology in the space between the two episodes or they are two different places all together. Atlantis, as seen in Neptune's Spatula, could simply be named after Atlantis in Atlantis SquarePantis and be where King Neptune resides as he rules over Bikini Bottom. Atlantis, as seen in Atlantis SquarePantis, is the mythical Atlantis itself and have no relation to Bikini Bottom at all. 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 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.
The Poseidome (where SpongeBob dueled against King Neptune to make thousands of patties)
The Royal Palace (Residence of L.R.H.)
Old Bubble Exhibition building
The Weapons Vault (contains all Atlantean-created weaponry, as the Atlanteans sealed away their entire arsenal to live in harmony with any sea life around them)
The Art Gallery (filled with paintings and other fine works of art)
The Money Mountains (contains all wealth the Atlanteans obtained, albeit the Atlanteans also sealed this massive stockpile away in a vault so that they may focus on knowledge)
Invention Center (the science center of Atlantis)
Dining Hall (a room in which visitors are given the best of Atlantean cuisine)
The Landing Gardens (a great courtyard near the Atlantean Royal Palace)
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.