If you use anything like Netflix or hulu plus that provide HD content, using a component cable will force the HD content to 480i. Unless you have a console that doesn't have an HDMI port, use an HDMI cable instead.
1080p does run with component, however like said before in other post make sure that your game you are playing supports you resolution size. I always run HDMI with auto in my counsel and it seems to run fine.
Component cables can support a 1080p signal but its dependant on whether your TV will accept it or not. Most TV's will go a max of 1080i through component. If possible look up your TV specs and see if it lists the resolutions the TV can display over the
The Xbox 360 does support 1080p output over component, but far from all TVs accept resolutions higher than 720p/1080i that way.
Beware though that only a handful of games support 1080p natively, most target 720p and get upscaled. The same goes for video apps - Zune supports 1080p natively, but the other apps are limited to 720p.
But the improvement over a standard composite connection is substantial either way.