Instructions
Exit out of whatever game you're playing and return to your Xbox 360's main menu (if applicable). If you're not playing a game, you don't have to do this.
Press the circular, "Xbox" button in the center of your Xbox 360 controller. This will bring up a menu of options on screen.
Select "Shut Down Console." This will power down your Xbox 360 and place it into "Standby" power saving mode. The only way your Xbox 360 will actually turn completely off at this point is if you flip the "On/Off" switch on the console and then unplug its power cable from a wall outlet.