1080 refers to a certain resolution of high definition content. What you need to consider is what resolution a certain size screen can offer you. Commonly, 32" - 40" sized panels had resolutions of 1366 x 768. As you get larger screens, this res becomes a little too small, and so image sharpness can suffer as a result. Some displays now have 1920 x 1080 resolutions instead.
720 and 1080 HD content refers to the vertical resolution of the source / display. So a 720 HD movie would have 720 vertical lines, and a 1080 would have 1080. As such, a 1366 x 768 res TV could naturally display 720 content, but because it lacks the full vertical resolution, it cannot truly show 1080 content. Instead, this content is downscaled by the TV's electronics.
the "i" and "p" refer to interlaced and progressive scan signals respectively. In basic terms, P is better. 720p sources are considered by many to be of similar quality to 1080i....1080p is obviously the main high res right now. Bare in mind many TV's are labelled as 1080 compliant, but often they just mean you can feed it a 1080i/p signal and it will accept it, but in fact the vertical res isn't there to support it being shown naturally. What you want, if you want a true 1080 LCD TV, is one which has 19200 x 1080 resolution AND can support 1080p (progressive scan) content.
Where this comes into play really in practice is when you look at larger screens. you wouldn't really want a 50" LCD for instance with only 1366 x 768 res (or 720p res), since it's not really enough. Likewise, there's probably not much need for a full 1080 support from a 26" LCD as you'd probably never really see the difference from 720 anyway.
Hope that helps
