what the market currently refers to as high definition (HD) content is typically:
1280x720 (720p) or 1920x1080 (1080p) in resolution.
therefore if you want to "fullscreen" high definition content you should be looking for a 1920x1080 monitor. granted you can "fit" the video feed to any size monitor at any aspect ratio but you might get black bars on the top/bottom/sides or a reduction in video quality depending on the exact scenario.
remember, panel type also matters. this will affect things such as viewing angle, color accuracy and how well gradients look.