Both HD DVD and Blu-ray store movie-based video on the discs in 1080p at 24 frames per second (fps). Some players do not output 1080p, but rather 1080i due to the circuitry implemented in the player for cost-savings or product maturity reasons.
Most displays are not capable of accepting a 1080p signal at 24 frames per second. More than likely, they need a 1080p signal at 60 frames per second (if they can take 1080p at all), so the conversion from 24 fps to 60 fps still needs to happen somewhere (this conversion is called 3:2, or 2:3, pulldown). The fact that most TV's cannot handle 1080p24 is why most HD DVD players and Blu-ray players alike output 1920x1080 at 60 fps in either interlaced or progressive fashion.
Does this mean that you're losing picture quality if your player is outputting 1080i, or your TV is only capable of accepting 1080i? As long as your TV is capable of displaying 1920x1080 pixels on the screen, and its internal circuitry was designed properly, there won't be any difference between a 1080i and 1080p input. The HDTV will deinterlace the 1080i signal, create a 1080p signal from it, and then display it to you. Unlike in the old days of analog video, there is no information lost or artifacts introduced as a result of the deinterlacing process. All the information is there, it just has to be reassembled.