This is the Gnome shutdown window. If you want to turn your computer off, choose "Shut Down" rather than "Reboot," then press "OK." Wait a few moments for your computer to prepare itself, then turn it off. If you see the line "System Halted," it's safe to power down the system. If your computer has Advanced Power Management support, like many laptops and some desktops, you may not need the power button.
To reboot, just choose "Reboot," press "OK," and wait.
Using the Advanced tab, you can change to any runlevel. There is also a preferences dialog which allows you to configure the shutdown command line. If you don't know what a runlevel is, or how to shut down from the command line, you should stick to the Easy tab. If you do know these things, the Advanced tab should be self-explanatory.