Most of the above info is correct if the TV is a standard def TV. However, you state that you have component plugs, and this leads me to believe you have an CRT HDTV. What's the make and model number?
Most (if not all) CRT HDTVs can accept 1080i, however, most CRT HDTVs cannot resolve more than 960 lines of resolution. Assuming that you could get the mini to display 1080i (with the help of SwitchResX or DisplayConfigX), you might be able to get decent results.
Now here's the catch. There's no easy way to get video out of a Mac using component cables. I've read where the Roku device (not the sound brigde, but the one that displays pictures) can be used to output through component at HD resolutions.
All in all, we need more info to help. Namely the model of the TV.
BTW, if you misspoke regarding the component connections, and only have composite connections, then yeah, it's not going to be pretty.