How can I set my computer so that I do not have to do a check in every time I open it or after it has been on for 10 minutes?

I worked with an Apple Support person about 1 1/2 hours ago and we thought that we had the problem solved but we did not. I cannot leave the computer open for more than 5-10 minutes without returning to the sign in screen. You know the one where you see the figure and your name and have to enter your apple password.

I have checked the books that I have and nothing has worked to remain signed in. Advise please.