What is the difference in 1080i and 1080p?

Both formats have a resolution of 1920 x 1080 pixels. The difference is in the way the picture is refreshed.
"i" stands for interlaced. The image is drawn one line at a time starting with all the odd numbered lines, 1, 3, 5 and so on to line1079. Then the image is completed by drawing all the even numbered lines from line 2 to line 1080. Each pass is known as a field and 1080i normally uses a field rate of 50Hz or 60Hz. A complete refresh is completed 25 or 30 times each second.

"p" stands for progressive scanning. In this case, the image is drawn during a single field. Lines are drawn consecutively starting with lines 1, 2, 3, through line 1079 and 1080. As each field is normally drawn 50 or 60 times each second so the image is fully refreshed 50 or 60 times each second. To display a 1080p image needs data at twice the rate of a 1080i image because lines are drawn at twice the rate.

Some comments about frame and field rates: In Europe, the field rate is 50Hz while in North America it is 60Hz, hence the two frequencies mentioned above. They are not the only frame and field rates however. Some other rates are used as well. For example, film is normally shot at 24 frames per second. It is quite possible to capture film to a 1080p format but at 24 frames per second. The full format is identified as 1080p 24. Based on the information above, you can see that the data rate for 1080p 24 will be less than 1080p 50 and less than 1080i 25.