The Tower of Babel was a tower built during the construction of the city of Babel, or which would later become the influential city of Babylon. The purpose of the Tower was for the people constructing Babel, to be able to give honor to themselves in order to keep themselves from scattering across the earth, planning for the tower to reach the skies . Though Noah and his family had been told to spread out, the people had stayed close for several years. Late in Noah's long life, his great grandson Nimrod was a leader in building several cities in the area between the Euphrates and Tigris river. In Hebrew, the names of Babel and Babylon (Greek spelling) are spelled the same, so it is to be assumed that the site for the tower was near the location of city that later went by that name.
The endeavor was in violation of God's command given to Noah and his family to fill the earth. As a result, He judged the descendants of Noah, still speaking his original language, by confusing the languages between the different family groups to make cooperation impossible. Nearby Ur would be the home of some of the descendants of Shem, probably including his son Arphaxad (spelling suggest Ur-phaxad). It was there that Abram (later named Abraham) would be born.
A united humanity made up of the generations following the Great Flood, who spoke a single language, were migrating from the east and came upon a large plain in the land of Shinar  It was here that they decided to stop migrating and build a city and a great tower as they did not want to be scattered across the face of the earth.
Previously God had commanded both Adam and Eve and Noah's family to reproduce and fill the earth . However, after several generations  humanity attempted to build a single, centralized location. Therefore God came down to the earth in order to confuse languages and make it difficult to collaborate amongst themselves. After the disruption in language humanity separated into individual groups scattering the Earth and setting up separate civilizations leaving the city to be temporarily abandoned.
Babel comes from the Hebrew word בבל (transliterated: Babel) which means "to jumble" or "a confused medley of sounds".
The phrase the "Tower of Babel" never actually appears in the Bible. The Tower itself is always referred to "the tower" or "a tower" throughout the passage. The city is named Babel because it is the location where God confused man's language. The city of Babel was later rebuilt as the city of Babylon.