720p is "progressive", while 1080i is "interlaced".
With a progressive picture, the entire picture is sent for each frame. With interlaced, only half the picture is sent at a time and the television blends it with the previous "field". So it's still 60 "fields" being sent each second, but it takes 2 fields to produce a complete frame (one field contains the odd lines of resolution and the other contains the even lines of resolution).
So with 1080i you get 30 frames per second (comprised of 60 interlaced fields). 720p doesn't mess around with interlacing...it just sends the entire frame all at once, hence 60 frames per second.
720p is better at producing smooth motion (great for sporting events, where there's a lot of fast motion). But, when you do the math, you get a lot less resolution with 720p vs. 1080i. There's a huge difference in pixel count between the two formats. 720p's resolution is 720 x 1280. That's 921,600 pixels. 1080i's resolution is 1080 x 1920. That's 2,073,600 pixels. There's more than twice the picture information in every frame with 1080i!
So, when watching stuff with mostly slower motion (sitcoms, late night talk shows, geriatric horse racing), 1080i looks twice as good.
Of course, 1080p blows them both away, as it has the same pixel count as 1080i (2,073,600 pixels) and doesn't mess around with interlacing. so you get the great smooth motion that 720p provides, plus the picture quality of 1080i.
The UK has their own formats. PAL 25 sends 25 frames instead of 30. I'm not sure why...maybe to reduce the bandwidth requirement.