What does Genesis 10:1 mean?
This verse introduces what is sometimes called the table of nations. In addition to listing some of the descendants of Noah's sons, the chapter also mentions the lands and nations that formed among these descendants. This is the purpose God intended for mankind in His commands after the flood: to repopulate the earth (Genesis 9:7). As part of His will, God had promised to never again destroy the earth with a flood (Genesis 9:11).It's an awesome thought to realize that the descendants of these individuals became the peoples of the ancient world, the nations that Israel would eventually interact with as a nation themselves. The incidents of the prior chapter have a drastic impact on these future generations. Since Ham dishonored Noah (Genesis 9:24), his son Canaan was cursed. The out-workings of that curse will not be clear until many generations later, when Israel arrives in the Promised Land.
The verse is clear that these sons were born to Noah's three sons after the flood and not before.