1080 and 720 are resolutions. So a 1080 TV will be much clearer than a 720 TV because there're more pixels, assuming the source video is 1080. 1080 is 1920x1080 pixels, 720 is 1280x720. In comparison, standard TV is 720x480.
Then the P on the end versus the I on the end. P is generally better.
P stands for progressive scan. Each line of each frame is drawn in order, from top to bottom one line at a time. Progressive scan usually looks more detailed, because people perceive the resolution to be greater, even if the resolutions are really the same.
The I stands for interlacing. Most standard tube TVs use this. The difference with interlacing is that it draws every other line at first, and then draws the rest of them. This image explains it:
Because of that, the picture doesn't appear as good as it would with a progressive scan picture.
The smaller the size of a TV, the sharper it will usually look. Assuming they have the same number of pixels, the smaller it is, the more pixels will be packed in there, so the sharper it will be.