It is broken into two groups
1080 is the resolution. On a computer monitor that would be 1920x1080, making the 1080 the number of pixels from the bottom of the screen to the top of the screen.
The i on 1080i stands for interlaced. A good article on interlaced is here: http://en.wikipedia.org/wiki/Interlaced
If it is p as in 1080p, that if progressive. Progressive is desired over interlaced especially for LCD screens. Here is an article that describes it: http://en.wikipedia.org/wiki/Progressive_scan
720i/p and 480i/p are resolutions also. This chart details when the increase in resolution matter:
I have a relatively small LCDTV at 32". I would have to be closer than 10 feet to notice an increase from 480p to 720p.
Many games on XBOX360 and PS3 do not display at 1080p. This is because doing so would emphasize graphical imperfections and possibly slow down the framerate.
