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.
No. The only Mobile Apple systems that came with built in Fm Radio capabilities were the Ipod Nano, 5th and 6th generation. What happened was, they placed the receiver/decoder into the Nano, and placed a micro antenna into the headphones that were sold at the time.
Fret not though, for you can still get radio on Ipod touch. If you would like the free version, then connect to WiFi (Wireless Fidelity...no clue why they named it this but...meh), go to the appstore, and download a free internet radio app. Unfortunately, this is limited, but still free.
If you don't mind paying a little bit, then you can purchase a radio accessory that you plug into the IPod that acts as both the receiver and antenna! In fact, most of these also double as an extra exntended life battery pack! How great is that?!