In Genesis 10:25 and 1 Chronicles 1:19 it notes that the earth was divided in the days of Peleg.

Gen 10:25  To Eber were born two sons: the name of one was Peleg, for in his days the earth was divided [palag]; and his brother’s name was Joktan.

1Ch 1:19  To Eber were born two sons: the name of one was Peleg, for in his days the earth was divided [palag]; and his brother’s name was Joktan.

Peleg was so named because of the division of the earth at that time.  According to Brown-Driver-Briggs, peleg means division.  The Hebrew word translated divided is palag which means to split or divide.  The Hebrew word palag appears in two other verses.

Job 38:25  “Who has divided [palag] a channel for the overflowing water, Or a path for the thunderbolt,
Job 38:26  To cause it to rain on a land where there is no one, A wilderness in which there is no man;

Psa 55:9  Destroy, O Lord, and divide [palag] their tongues, For I have seen violence and strife in the city.

The usage of palag in these verses show that the word has a broad application, from geology to language.

But what was the nature of the division that occurred in those days?  The chapter following on from Genesis 10:25 gives the answer.  Genesis chapter 11 describes the Tower of Babel.  Mankind built the Tower of Babel so as not to be scattered. 

Gen 11:4  And they said, “Come, let us build ourselves a city, and a tower whose top is in the heavens; let us make a name for ourselves, lest we be scattered abroad over the face of the whole earth.”

This displeased God because He had instructed mankind to fill the earth and mankind was now deliberately going against His instruction.

Gen 9:1  So God blessed Noah and his sons, and said to them: “Be fruitful and multiply, and fill the earth.

So, God confused their language to force mankind to scatter and thus fill the earth.

Gen 11:7  Come, let Us go down and there confuse their language, that they may not understand one another’s speech.”
Gen 11:8  So the LORD scattered them abroad from there over the face of all the earth, and they ceased building the city.
Gen 11:9  Therefore its name is called Babel, because there the LORD confused the language of all the earth; and from there the LORD scattered them abroad over the face of all the earth.

Directly following the account of the Tower of Babel is the genealogy of Shem.  Peleg is listed in Shem’s genealogy. 

Gen 11:10  This is the genealogy of Shem: Shem was one hundred years old, and begot Arphaxad two years after the flood.
Gen 11:11  After he begot Arphaxad, Shem lived five hundred years, and begot sons and daughters.
Gen 11:12  Arphaxad lived thirty-five years, and begot Salah.
Gen 11:13  After he begot Salah, Arphaxad lived four hundred and three years, and begot sons and daughters.
Gen 11:14  Salah lived thirty years, and begot Eber.
Gen 11:15  After he begot Eber, Salah lived four hundred and three years, and begot sons and daughters.
Gen 11:16  Eber lived thirty-four years, and begot Peleg.

Adding the years in bold (above) we see Peleg was born 101 years after the flood.

With this background we now can understand the course of events.  After the flood God instructed mankind to fill the earth.  Over the next 100 years mankind refused to fill the earth but instead built the Tower of Babel to prevent themselves from being scattered over the face of the earth.  God intervened by confusing their language thus forcing mankind to scatter and fulfill His purpose.  Peleg was born when the scattering was happening which was why his parents gave him his name.  This scattering occurred in the days of Peleg. 

In what way was the earth divided in the days of Peleg?  God scattered mankind over the face of the earth by confusing their language.