In the first Virtual Families (VF1), when there are no surviving children to begin a new generation it's the end of the game and you'll have to start over from the very beginning.

In Virtual Families 2 (VF2), when there are no surviving children to begin a new generation, you'll get a message that your family is lost, and that you need to choose someone new. Click "Ok" and you'll be taken to the adoption screen where you choose a new person to begin a new family lineage. The new person will take over the house, all of it's contents, the Bank account, and the completed goals. They just won't be related to the previous families, and there will be a "break" in the family tree.