Does the iPod with video have a built in FM radio? I have an iPod with video and in the Settings there's an option to show "Radio". However, when I tell it to show, it doesn't show up in the home screen. Does it have a built-in FM radio?
The Radio option probably refers to an accessory that Apple used to sell, the iPod Radio Remote.

Your iPod does not have an FM radio built in; only the 5th and 6th Gen Nanos have that.

For more information about using the iPod Radio Remote, see this Apple KB article.