Here are your most likely options, from the best to the worst, if you want to hook up your desktop PC or notebook to your TV.
HDMI to HDMI: If both devices have HDMI connectors, you're in luck. One simple cable will give the best picture and sound that your TV can handle.
If you don't get any sound, you may have to change Windows' audio output device settings. In Windows 7 or Vista, right-click the speaker icon in the system tray and select Sounds. Click the Playback tab and select Digital Audio (HDMI).
DVI to HDMI: If your computer lacks an HDMI connector, it might still have PCI. With the right adapter or cable, this will work just fine. You can pick up DVI to HDMI adapters for less than $10.
These two standards send and receive identical digital video streams, making them easily compatible. The only differences: DVI carries no audio signal, and the connector looks funny.
The adapter takes care of the funny-looking connector, but you probably can't get sound from your computer to your HDTV if you're going over DVI. The television was built on the assumption that an HDMI signal would carry its own audio, so there may be no audio inputs associated with the HDMI connectors.
The solution is to bypass the TV speakers. If you have a separate receiver (a good idea in a home theater), plug your PC's audio output into that. You may need a Y-adapter audio cable with a one-eighth-inch stereo mini plug (like the one used by your headphones or earbuds) on one end, and two standard RCA audio connectors on the other. Or you can set up some computer speakers.
VGA to VGA: If your PC lacks HDMI and PCI outputs, it almost certainly has a VGA connector. And I have yet to see an HDTV without a VGA port (which it may identify as RGB, PC, D-sub, or RGB PC). VGA can send any signal your PC and HDTV can handle.
At least, it can if you get it to work properly. Getting a good VGA signal on a television involves tweaking settings on both the PC and the TV. Start by reading the VGA section of your HDTV's manual.
How successful you'll be, and how much work that success requires, depends on your TV. My experience is that newer sets are pretty much plug-and-play in their VGA connection, but older ones give you grief. Some simply cannot give you a decent picture.
Not all HDTVs supply an audio input associated with the VGA connector. If yours doesn't, see the last paragraph of the PCI to HDMI section above.
S-Video to S-Video: If both devices have an S-Video connector, and they share nothing better, buy an S-Video cable and connect them. Then use a Y-adapter audio cable to connect your PC's audio to the audio inputs associated with the TV's S-Video connector.
This is a strictly analog, standard definition setup. Your TV will get nothing digital, and nothing HD. But it's simple and it works.
None of the above: If none of these matches work for you, you're out of luck. You're not going to get that particular PC working with that particular TV. Sure, you can buy a VGA to S-Video connector; I've tried several. I've yet to find one that worked.