The 1080 and 720 refer to the vertical resolution used - standard TVs I believe are 625.The i refers to interlaced - the same system that standard PAL uses where on one pass every other line is done, eg lines 1,3,5,7 and then the second pass lines 2,4,6,8...etc are done.
The p is for progressive - where all the lines are done every pass, giving a crisper, clearer picture.