If you and your family member will use the same computer, you simply open TurboTax and 'start a new return' for the other family member. Then when you open TurboTax, you have two returns showing that can be worked on.
If the family member wants to use a different computer, they will need to login to your Intuit account and download the program to their computer (or you could login on their computer and download it for them). You can have the program on five of your own computers.
Here's more info on Installing TurboTax on Multiple Computers.