What effect does feeling Well Rested have?
Well Rested is a temporary perk that gives you a 10% XP bonus for a period of time. The wiki lists this period as being 12 hours. You can see the perk on the perks page in your Pip-Boy.
Source
To add on to the previous answer, You have to sleep in a bed that you own for one hour and you will get the +10% experience for 8 hours in game.