720p, 1080i and 1080p are shorthand name for a category of video modes.
720p is a smaller HD resolution than 1080p. 720p is made up of 1280 (horizontal) x 720 (vertical) pixels. 1080p is made up of 1920 (horizontal) x 1080 (vertical) or 1440 (horizontal) x 1080 (vertical) pixels in 1080i.
In other words, if you compare 720p and 1080p movies on a 42" TV, you will notice that the 1080p most likely looks better. The pixel size in 720p is much larger on a 42" TV than 1080p. The smaller the pixel size the sharper the picture is.
WikiPedia 720p - WikiPedia 1080p - WikiPedia 1080i
Unfortunately, to play these high resolution, high nitrate releases you will need a relatively powerful cpu and a few pieces of software that you might not already have. To help make your life easier, I am going to walk you through it.
First you need to have a video player of any kind. I like Videolan as it play nearly everything and do not contain any DRM features but you can also use media player classic, which you can grab from here.
Then you need to download and install the Combined Community Codec Pack, which you can download here:
The CCCP (Combined Community Codec Pack) is a DirectShow filter pack specifically built for playing anime. It has been assembled by some of the most knowledgeable people in online anime distribution, and one of the primary design goals is to play as much as possible with as little as possible. The CCCP attempts to be as unintrusive on your system as possible. [WIKI]
If you find your hardware can't play it quite quick enough with CCCP, You can still try to buy a copy of CoreAVC, the fastest H264 software decoder (but you may still need some of the software from CCCP, so keep that on there).
Last but not least you should have AC3Filter installed, which you can find right here:
AC3Filter is a high quality free audio decoder and processor filter.
It allows media players to playback movies with AC3 and DTS audio tracks. Also it can process any audio track, much of processing options allows to adjust the sound in almost any way. It is possible to up mix any audio source to 6 channels. Filter can do multi-channel and digital (SPDIF) output. It can encode any audio source to AC3 on-the-fly and send it over SPDIF to the receiver.
Even by installing all these codecs, and tools, there is no guarantee that you'll be able to enjoy HD 1080p playback on your PC. 720p is a lot more easier to play on any couple of PC/graphic card. Having a PC sitting near the television may also not be an option for You. In that case, you may be more satisfied by a hardware box: PS3, XBOX360, dedicated boxes (TIVO)