Trantor is depicted as the former capital of the First Galactic Empire. Its land surface of 200 million sq. km. was, with the exception of the Imperial Palace, entirely enclosed in artificial domes. It consisted of an enormous megalopolis that stretched deep underground and was home to a population of 40,000,000,000 (40 billion) human inhabitants.
Trantor was first described in the 1940s when the Foundation Series first appeared in print (in the form of short stories). Asimov described Trantor as being in the center of the galaxy. In later stories he acknowledged the growth in astronomical knowledge by placing it as close to the galactic center as was compatible with human habitability. The first time it was acknowledged in novel form was in Pebble in the Sky.
Trantor represents several different aspects of civilization. At once it is both the center of power in the Galaxy, and also the administrative head. It is also an illustration of what could eventually happen to any urbanized planet. Asimov used the Roman Empire as the creative basis for the Foundation series, so Trantor is in some sense based on Rome at the height of the Roman Empire. Trantor also illustrates the mentality of human beings that was first encountered in Isaac Asimov's The Caves of Steel, wherein human technology will ultimately result in a complete encapsulation of a population, and that population will eventually suffer psychosis associated with that total encapsulation. Asimov did once say that these encapsulated cities represented the kind of place he'd like to live. He did not even realize how distasteful some people found this until someone asked him about it.
There have been some serious attempts to illustrate a planet like Trantor in the Star Wars films by George Lucas, the first being the Death Star and the other being Coruscant. The Death Star isn't a city as such since it is entirely man-made (it's more like "Roger's Planetoid" in E. E. Smith's Lensman series). Thus, Coruscant is one of the more convincing images on screen we have today of Isaac Asimov's conception of the world-girdling city of Trantor.