Most of the time, you will never need to manually restart your iPod nano: iTunes will take care of it whenever it applies an update; and your nano automatically goes to sleep to save battery life whenever you do not use it. But if you start having a few problems, like a blank screen, or no sound, or if your iPod nano becomes unresponsive, restarting it often fixes basic problems.
Before you restart your iPod nano, make sure that it is not currently plugged into your computer (Mac or PC), or at least that it is not currently being synced with iTunes; a seemingly unresponsive iPod can often simply be in mid synchronization, or just being recognized by your computer if you just restarted it.
To restart your iPod nano, click on the Center button, and hold down both the Center button and the menu button (top of Click Wheel) at the same time, and held down for about 5-6 seconds: if you look at the screen, you will notice that at that point, the screen becomes blank (black), and remains inactive for a few seconds. This is your visual confirmation that your iPod nano is currently in the process of restarting, and you should within a few seconds see the Apple logo appear on screen.
iPod nano Tip: Unlike the iPod touch, you do not need to use the Hold button (Sleep / Wake button on the touch) at the top of your nano to restart it (restarting your iPod touch requires holding both the Home and Sleep buttons - but not so with the nano - the top button is only used to lock the click wheel menu).
When your iPod nano's screen freezes (and the iPod is not being synced by iTunes, or connected to a computer in the process of restarting itself), it is precisely the time when a manual restart is likely to help: much like your computer's power button, holding down the Center button for 5 seconds ignores the "software" aspect of your iPod, and will restart it.