Those remaining on the earth during the Millennium will be the good people of the earth—those who were righteous and just, regardless of their religion. This includes those who will be on the earth at the time of the Second Coming as well as those who will be resurrected at that time and other times throughout the Millennium. The wicked will not remain on the earth and will not be resurrected until after the Millennium.
So, what will people do during these 1,000 years? We know that a main focus will be to continue the work of salvation—helping people come unto Christ. Church members will share the gospel with those remaining who have not received it in its fulness. Family history and temple work will be especially important and will link the entire family of man all the way back to Adam and Eve.
A thousand years is a long time, so it’s safe to assume that people will likely do a wide variety of things that are “virtuous, lovely, or of good report or praiseworthy” (Articles of Faith 1:13) but that have not necessarily been mentioned in any of the revelations. For more on the Millennium, see True to the Faith: A Gospel Reference (2004), 103–4.