When a set shows you what mode it's in, it's telling you the number of scan lines being used, which is 720 or 1080. (Could be as low as 480 if you're viewing a non-HD picture.) The other number is irrelevant, as it's the horizontal resolution. It's there. If you're looking at a 1080 picture, you've got the 1920 across. If you're looking at a 720 picture, then it's 1280 across.
The rest is optional, explaining the reason for p or i at the 1080 resolution.
Do the math: a 1280x720 picture is just under a megapixel. A 1920x1080 picture is just under 2 megapixels. The digital broadcast standard only leaves room, with reasonable compression, for about a megapixel every 60th of a second, so a 1920x1080 picture can't be drawn in the time allotted. That "how-much-how-fast" issue is called bandwidth.
To broadcast a 1080 picture, they break it into two parts, alternating all the even lines in one 1/60th-second period, and then the odd lines in the next. The set mixes them together to give you a 1920x1080 picture every 30th of a second. (The times may be different where you are, like 1/50th and 1/25th.) That odd/even mix is called interlacing, and it allows a picture to be created that's more information than the bandwidth allows, by sacrificing frame rate in favor of resolution. That's what they mean by 1080i. The 720-line picture fits completely in a single 1/60th second time span, so it's scanned progressively, all lines in one sweep, for 720p.
Your Blu-ray player and game console do not have the broadcast bandwidth limitation, so it can fit the whole 2-megapixel picture in a single 1/60th second scan, so it gets 1920x1080 progressively, for 1080p.
