ANSWER: The resolution your television displays has to do with three things:
1) The television station sending the signal
2) The tuner decoding it
3) The capability of your TV to display that signal
High definition televisions have the ability to display resolutions of 720p, 1080, or 1080p depending on the model. That means they can display high definition content as long as the signal is in HD and the tuner is able to decode it - meaning an ATSC HD tuner. Tuners are either built-in to the television or come in an external receiver or set-top box usually provided by your cable/satellite company. Though, you can buy them at any consumer electronics store like Best Buy, Circuit City, Frys, etc.
Television Stations Control The Signal
Television stations are in control of what kind of signal we get. Right now, some stations send out a digital and analog signal. Within the digital signal they embed a high definition feed that is usually offered by way of a separate HD-only channel, which when they aren't showing HD programming the screen is configured with bars on the side to give off a higher resolution 4:3 aspect ratio. Most of the stations with two signals are OTA, over-the-air, broadcast networks like CBS, ABC, NBC, UPN, etc. However, some cable stations offer a HD channel for an extra monthly charge to your cable/satellite bill.
Cable/Satellite Programming
As for cable/satellite programming, the non-HD digital service is nothing more than 480i standard definition. Your TV will not, can not, reconfigure that signal to enhanced or high definition. You are confined to whatever signal they send out. Keep in mind, they may send out an alternate HD feed on their HD channel. So in a sense, your TV has nothing to do with choosing the best signal available, but on the other hand, without a digital tuner you would not be able to get the best of digital programming.
All Digital Signal
In a few years when the analog feed stops, you'll probably experience one channel per station like it was years ago. There won't be a need for a separate analog channel because analog signals won't exist within the television spectrum. This is where your question is most valid as your HDTV, in conjunction with the digital tuner, will be the deciding factor in receiving and displaying the best signal possible. It will do this automatically because the TV stations will only send one of two feeds, standard or high definition. Your tuner will then decide which one is best for your TV.
DVD Resolution
As for DVDs, they only display in a 480p resolution because that's the only resolution available for DVDs at the present time. Even if your TV is set at 720p, 1080i, or 1080p, your DVD will still playback at 480p.
How To Display The Best Resolution
If you're concerned with missing out on the best resolution, set your television to display the highest signal possible that's compatible with your tuner. Then you won't have to worry anymore because you know your TV is ready to display the best of signals.
