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The struldbrugs are totally unique to Luggnagg. Gulliver is introduced to this term by "a person of quality" (3.10.2). (Gulliver loves speaking to people of quality, i.e., people of the upper class. Snob!) This unnamed person of quality asks Gulliver what he would do if he could live forever? Gulliver really likes this idea. He says he would make tons of money, learn everything there is to know in the world, and then spend all of his time talking to other immortals, who must be equally as brilliant as he would be.
The person of quality laughs at Gulliver's stupidity. The thing is, Luggnagg has immortals, the struldbrugs. Perfectly ordinary parents can have them, and they are marked by a dot above the left eyebrow that changes color as they grow older until it hits black at age forty. Oddly, the struldbrugs tend to have normal, mortal children.
These immortals, unlike our fantasies of, say, Edward Cullen in Twilight or Vampires Bill and Eric in True Blood, are not eternally young. They age normally up until age eighty, when, the Luggnaggians imply, most decent people have the sense to die. Once the struldbrugs hit 80 years old, they have "not only all the follies and infirmities of other old men" (3.10.13), but they are also extra-opinionated and cranky because they're worried about living forever.
Thus, the struldbrugs provide a satire of both old age and the dream of living forever. Gulliver's description of their decayed physical condition is pretty horrible, but their bodies aren't even the worst problems the struldbrugs face. The problems with living forever as an old person include:
The poor struldbrugs poke fun at the kind of fantasies of immortality that Gulliver starts with. He imagines that he would hold basically all the power and knowledge in the world thanks to his long life. Really, becoming hugely wealthy and learning everything would mean depriving future generations of their own opportunities to own land and invent new stuff. Gulliver is dreaming that, if he lives forever, the world would stagnate around him and nothing would ever change – the world would be fixed in one place to suit him. That's a hugely vain, stupid thing to wish: the world will never stand still for one person, no matter how long he lives.