They also look better than they do on my friends plasma TV which is supposed to be "high definition". That could come from the fact that my monitor is a lot small than his TV, so the picture is kind of condensed... if that makes sense.

Anyway I do know that LCDs are lighter and less expensive, plus You could probably hook your computer up to it for the ultimate flight sim view!
High definition merely refers to the number of pixels going up the side of the screen. 720 means 720 vertical pixels, 1080 is 1080 pixels. Depending on where you live, is anywhere from between 480 and 576 p/i. Of course, if the media you're looking at is standard definition then, as you said, the image will be blown up and not look as good. High Definition doesn't mean the image will look good, only that a certain number of pixels are present. The same goes that you shouldn't base the quality of a digital camera off the number of pixels (mega pixels, to be more accurate) that the photos have. All that says is how big the image is, not how good the image is. My little analogy is this: Imagine you have an A4 sheet of paper and an A1 sheet of paper... Give the A4 sheet to (an alive) Salvador Dali and give the A1 sheet to an infant... The chances are, the image by Dali will be far better though the one by the infant will be bigger. So it goes to show that size isn't everything and it is certainly not a show of quality, though as size increases,
usually the quality gets better too.
For the quality of image, check the contrast ratio figure to show you how intense the range is from white to black on the screen which will give you nicer looking images. And of course, if you can, see a running demo of the TV to get an idea of its actual quality. A lot of shops show the TVs working.
The 1080i or 1080p bit tells you how the image is displayed (interlaced or progressive). I believe progressive is where it loads the image from top to bottom going down each pixel row, and interlaced is where it loads every odd row then the even rows. Progressive gives less blur and slightly better frame rate.
If you're sitting far back or plan to invest in HD-DVD (pretty much dead technology) or Blu-Ray, then get HD. If you have a TV service providing HD channels and they are ones you'd be likely to watch, get HD.
An LCD or Plasma should connect to your computer easily enough. My near-decade old CRT TV (which we still use as it does the job just fine) can run the video from my computer if I wanted it to.
I prefer the picture on LCD, it seems much clearer.