In order to watch online (Internet) content on your TV, all you need is a video cable running from your PC's Video Card out to an input on your TV. Your TV will then act just like a computer monitor. What cable you need depends on what type of video card output your new PC has (VGA, DVI, HDMI or even S-Video) and what type if video input your TV has (VGA, DVI, HDMI, S-Video, Composite, Component or even just Coax).
If your PC's video output does not match your TV's video input, there are adapters and converter cables you can purchase (DVI --> HDMI, DVI --> VGA, VGA --> Component, VGA to Composite, and so on...). It all really depends on your hardware.
In order to record content from the cable provider, you will need an extra component in your PC; either a TV Tuner or a Video Capture Card. A TV Tuner Card will act very much like your Cable Box in that it will take the raw signal from your cable outlet and feed it into your TV Tuner Card. You can then use the TV Tuner Card in your PC to change channels and set up recordings of your favorite shows.
The down side to this is that you can only pick up what ever channels your cable company sends "in the clear", meaning unscrambled. Premium channels like HBO and Showtime are always scrambled as are many of the regular channels your cable provider offers (such as ESPN, MTV, and others). It varies from cable company to cable company.
The way to get around this is to connect your incoming video signal from the cable outlet to the cable box. Run the output from the cable box to the input of the TV Tuner. Run the graphics card output to the video input on your TV. This essentially turns your TV Tuner card into nothing more than a fancy VCR for your computer. You'll still use your cable box remote control to change channels, but you now get all the channels you pay the cable company for (and you can record them since the signal is descrambled by the cable box).
The software you need is usually included with the purchase of a TV Tuner card, but Windows XP Media Center Edition (MCE), Windows Vista Home Premium, and Windows Vista Ultimate all come with the necessary software as well. Vista Home Basic and Vista Business do not. Many people, myself included, prefer the Windows Media Center software this purpose.
If you do go this route and use Windows Media Center, look for a TV Tuner that includes a Media Center Remote Control. I personally find the Media Center Remote Control essential for browsing through my list of mp3 and movies quickly and easily. You may also want to consider a wireless keyboard. The wireless keyboard will allow you to login, browse the web, whatever from the comfort of your couch. I'd also recommend one with an On/Off switch to save on batteries.
I think that about covers it. If you have any other questions, feel free to ask.
-Wolf sends
Edit: One more thing. Make sure that the TV Tuner Card you get has an ATSC tuner at the very least. With the switch over to digital broadcasting come Feb of next year, you will need an Digital (ATSC) TV Tuner card.