The 'Table of Nations' is actually built from two separate genealogical sources. These can be easily distinguished by the way the text alternates between two verbal patterns in how offspring are introduced. When split apart, we have a mostly 'Completed' table on one side, and a 'Fragmented' table on the other. It is apparent that the latter table was inserted into the former as a supplement; it would not have been fragmented originally, but only because the entire thing was not included.
The Completed table is totally deadpan. It names Noah's sons, and merely lists out their respective descendants. Here, the names of Noah's sons anticipate regions of the world (relative to the land of Israel), and each of their sons personifies a nation from that region. The sons of Japheth (enlarge, spacious) represent lands to the north of Israel. The sons of Ham (hot, burnt) settle southward of Israel, especially in Africa. And the sons of Shem (renown, name) settle in the Near East, where the bulk of the biblical narrative takes place.
This first table has Noah's sons' sons spreading across the earth of their own volition, and languages naturally changing from their dispersion. This is contrary to the Tower of Babel story told in Genesis 11, the next chapter, where humanity is united in location and language until God intervenes.
The Fragmented table has a bit more of a story-telling approach. Instead of flatly listing out descendants, it pads that information by telling the reader a bit about some of the individuals named, such as the tribes which came from Canaan. The Philistines are said to come from the Egyptians. (Historically, the Philistines were from a group the Egyptians called the 'Sea Peoples'; surviving Philistine artifacts show they are related to the Greeks.)
Both tables agree with the genealogy in Genesis 11 that Shem had a son named Arphachshad. However, while the Fragmented table follows only Shem's son Arphachshad and his first few generations of descendants, the Completed table is more immediately concerned with the descendants of Aram, one of Shem's five other sons
Most tantalizing in the Fragmented table is the brief story told about Nimrod. The narration says 'he was the first on earth to become a mighty warrior' (contradicting the earlier story of the Giants in Genesis 6.1-4), as well as 'a mighty hunter before Yahweh'. This phrase is a point of debate, with one side claiming 'before Yahweh' means 'in defiance of Yahweh', while the other side says Nimrod was simply recognized by Yahweh as a great hunter.
In any case, Nimrod is credited with founding the first kingdom. The land 'Shinar' corresponds roughly to Mesopotamia, with the cities he founds being the major cities of the Sumerian, Assyrian, and Babylonian empires. Consequently, he is frequently connected to (and blamed for) the building of the Tower of Babel, though the text of Genesis gives no indication he was involved.
Speculation abounds as to whether the Fragmented table, or Nimrod's story in particular, come from the J source. The narrative style for Nimrod's story is similar to J's, and the note regarding Peleg fits with J's overtly symbolic naming scheme. In this regard, it is further speculated whether the text has been rearranged from its original position: Nimrod may belong before the flood with the Giants who were 'warriors of old, men of renown', or the Fragmented table altogether may come after the Tower of Babel story, since Nimrod's story presumes a non-centralized population. (Another possibility is that the Fragmented table, and Nimrod's story with it, are from another source entirely, unrelated to J.)
Completed // Fragmented
These are the descendants of Noah's sons, Shem, Ham, and Japheth; children were born to them after the flood.
The descendants of Japheth: Gomer, Magog, Media, Greece, Tubal, Meshech, and Tiras. The descendants of Gomer: Ashkenaz, Riphath, and Togarmah. The descendants of Greece: Elishah, Tarshish, Kittim, and Rodanim. From these the coastland peoples spread. These are the descendants of Japheth in their lands, with their own language, by their families, in their nations.
The descendants of Ham: Cush, Egypt, Put, and Canaan. The descendants of Cush: Seba, Havilah, Sabtah, Raamah, and Sabteca. The descendants of Raamah: Sheba and Dedan. These are the descendants of Ham, by their families, their languages, their lands, and their nations.
The descendants of Shem: Elam, Asshur, Arpachshad, Lud, and Aram. The descendants of Aram: Uz, Hul, Gether, and Mash. These are the descendants of Shem, by their families, their languages, their lands, and their nations.
These are the families of Noah's sons, according to their genealogies, in their nations; and from these the nations spread abroad on the earth after the flood.
Cush became the father of Nimrod; he was the first on earth to become a mighty warrior. He was a mighty hunter before Yahweh; therefore it is said, 'Like Nimrod a mighty hunter before Yahweh.' The beginning of his kingdom was Babylon, Erech, and Akkad, all of them in the land of Shinar. From that land he went into Assyria, and built Nineveh, Rehoboth-ir, Calah, and Resen between Nineveh and Calah; that is the great city. […]
Egypt became the father of Ludim, Anamim, Lehabim, Naphtuhim, Pathrusim, Casluhim, and Caphtorim, from which the Philistines come. […]
Canaan became the father of Sidon his firstborn, and Heth, and the Jebusites, the Amorites, the Girgashites, the Hivites, the Arkites, the Sinites, the Arvadites, the Zemarites, and the Hamathites. Afterwards the families of the Canaanites spread abroad. And the territory of the Canaanites extended from Sidon, in the direction of Gerar, as far as Gaza, and in the direction of Sodom, Gomorrah, Admah, and Zeboiim, as far as Lasha.
To Shem also, the father of all the children of Eber, the elder brother of Japheth, children were born. […]
Arpachshad became the father of Shelah; and Shelah became the father of Eber. To Eber were born two sons: the name of one was Peleg, for in his days the earth was divided, and his brother's name was Joktan. Joktan became the father of Almodad, Sheleph, Hazarmaveth, Jerah, Hadoram, Uzal, Diklah, Obal, Abimael, Sheba, Ophir, Havilah, and Jobab; all these were the descendants of Joktan. The territory in which they lived extended from Mesha in the direction of Sephar, the hill country of the east.