Whenever a conversation about HD and the 360 comes up, I see a ton of confusion about which outputs give you 1080p. Hopefully this handy chart from DanielN will help to answer a few questions.

Xbox 360 connected using component
DVD’s via HD-DVD drive:  480p
DVD’s via DVD drive: 480p
HD-DVD’s: 1080i
Games: 1080p
MCE Extender: 1080p
MCE Extender for OCUR PC: 1080P

Xbox 360 connected using VGA
DVD’s via HD-DVD drive: 1080p
DVD’s via DVD drive: 1080p
HD-DVD’s: 1080p
Games: 1080p
MCE Extender: 1080p
MCE Extender for OCUR PC: 520P

Xbox 360 connected using HDMI
DVD’s via HD-DVD drive: 1080p
DVD’s via DVD drive: 1080p
HD-DVD’s: 1080p
Games: 1080p
MCE Extender: 1080p
MCE Extender for OCUR PC: 1080P

The only other thing you need to check is what resolutions your TV supports on each input.

[Update] Thanks to MattG for pointing out that the new Premium boxes contain HDMI as well.