As an example, I have a Hauppauge card in my PC, and it comes with software called "WinTV". WinTV has a record button that will save your file in a standard video format, and that can be loaded into whatever mixing software you like (Windows Movie Maker is mentioned twice above). And finally, as Dan mentioned, Hauppauge makes an external version that connects via USB. I haven't used one of those, but I would imagine it would work just fine. The only reason I'm suggesting Hauppauge is because I'm familiar with the software that comes with it, and it works really well. Other brands may too, I just am not familiar with them.
I actually have two Hauppauge cards...both used to watch TV. One here in my office on my main computer, and one in the garage man cave. The one in the garage just has an antenna hooked up, and it gets a pretty good selection of HD channels over the air. The one in here is connected to a cable box.
But also on the one in the garage...I use it to test video game consoles. And it would be very easy to record game footage.