First there’s resolution. The pixels are arranged in lines across the screen. The resolution number 480, 720 or 1080 is the number of lines of pixels on the screen. Clearly, a 1080p TV will display more picture information than 720p. More information = a better picture.
To illustrate the difference:
On a 42-inch analog NTSC TV (the old traditional TV you have at home now), there are a total of about 200,000 pixels. A 42-inch HDTV has over 2 million pixels. That’s 10 times the information. Plus, to fit more pixels on the same 42-inch screen, they have to be smaller. Smaller, more tightly spaced pixels create a smoother, more detailed picture. And you can see the difference. (On some NTSC big screens you can actually see the lines of larger pixels.)