The Laputians are a race of weirdos whose heads are always leaning to the right or left and whose eyes never focus on the world around them. They live on a floating island controlled by a central magnet. The fact that they float through the air without direct ties to the Earth is pretty symbolic. These guys love two things: math and music. In both of these, they are very far advanced, but you know what they stink at? Anything practical. They can't build houses with right angles, and they can't sew clothes that fit. The reason all of their designs fall apart is because they refuse to take measurements from real life, preferring instead to use equations to prove what has to be true.
Still, despite the fact that Laputa floats, it does have political connections to the continent below it, Balnibarbi. All of the Laputian King's ministers have their estates on the continent, so the King can't just enslave the people living under their island. But the King does maintain a strict tribute policy. If the people below Laputa do not send their tributes, well, there will be hell to pay. The King will float his island right over their heads, blocking the sun and rain and dropping stones on them until they pay up. So you can see, even though the only thing these people really know is music and math, they still like to dabble in world domination – just human nature coming through, yet again.
Laputians are so distracted with their internal world of Deep Thoughts that they need to be reminded of what they are doing at any given time. That's why all the Laputian nobility must be accompanied by a servant at all times. These servants carry a kind of rattle at the end of a long stick that they use to touch the mouth of a Laputian who is supposed to be speaking, the ear of a Laputian who is supposed to be listening, and the eye of a Laputian who is supposed to be looking at something.
Because Laputian men spend so much time wrapped up in their own heads, they don't have much time for their bodies, if you know what we mean, and we imagine that you do. They're not big in the love department. So when men from the continent come to visit Laputa on business, Laputian women are totally willing to have affairs left, right, and center – they love strangers. And their husbands never notice, after all. They're too busy with their math and music to notice their wives stepping out on them. This is one possible explanation for the origin of the name "Laputa," from the Italian la puta, "whore." (For more on Swift and women, check out our theme on "Gender.")
The Lagado Royal Academy is a lampoon of the Royal Academy in London in Swift's day (Lagado being the main city of Balnibarbi). The experiments Gulliver records – men trying to turn poo back into food, extract sunbeams from cucumbers and so on – were real things that eighteenth century scientists tried to do (source: Robert Greenberg, Editor, Gulliver's Travels. New York: Norton, 1961, 133). It sounds like there were some pretty ripe candidates for the Ig Nobel awards hanging around back then.