Watching video is not the same as manipulating and processing the creation thereof. On your 7-year-old computer, you would have problems manipulating HD video, as it may not have the power to process it very well. The RAM on the video card is used to buffer the information that the card is processing, in the case of video production, when showing a preview in your NLE or rendering the video that you've created in your NLE or compositing software. The amount of RAM determines how much data can be buffered at any given time. Obviously, the more you have, the more data that can be buffered, and the faster it can be processed. The speed of the video RAM is also an important factor in that equation.
As far as a minimum of RAM needed to edit HD video, that's a tough question to answer, as different applications will have differing requirements, and it also depends on what you're trying to accomplish in the video. The fact is that your computer may be able to render a video, but getting your project to a point where it is ready to be rendered can be cumbersome, if not nigh impossible, on an older older computer that doesn't have much processing power, as you need to be able to preview the video in your NLE in order to edit it. Without some power under the hood, your video preview may look like a sloppy mess of several-second-long still pictures instead of an actual video. I would say that 2GB of video RAM is a healthy number that should cover most applications.
I hope I've been helpful, Joe. Let me know if you have any other questions.