HD output is implemented on many products by graphics cards manufacturers. Here is a table to determine if your graphics hardware supports HD video or not.

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The table above shows the graphics solutions claimed to support HD by their manufacturers. Intel's graphic products, for instance, are a very poor choice.
There are also many products that can play HD content well, even if it hasn't been claimed explicitly by the Manufacturer. If you have an older graphics card -say an NVidia GeForce 7200- you may still be able to play HD content. I was playing HD Ready content (720p) without any problems with my NVidia 7200 graphics adapter and an AMD XP 2600 processor. Trial & error is your friend here: you can just download a sample of 1080p content from the Internet - you do not need to go for a whole movie, just select a preview - and check if your computer plays it smoothly. If you have a Pentium IV 3.0 Gigahertz or newer CPU to help out your older graphics card, you might be able to play HD content without any problems. I emphasize newer, because if you have a Core 2 Duo running at 2.4 Gigahertz, which is newer than Pentium IV, but with a slower clock than the 3.0 Gigahertz, that's an improvement, despite the smaller number.