This is a classic question. In this post I will try to give you the facts about both so that you can decide for yourself:
Put simply, 1080p is works the same way that a picture in the movie theater works. In case you aren’t familiar with what I mean, I will explain. The “p” in 1080p stands for “progressive.” Progressive means that every frame is replaced by another “complete” frame. Complete simply means the whole screen is filled by the picture (i.e. not 50 percent of the screen.) Therefore, 1080p is “progressive,” because the motion is progressing through a series of pictures.
In 1080i, the “i” stands for “interlaced.” That means that only 50 percent of the screen is filled for each frame. As you probably already know, on a TV screen there are horizontal display lines, and vertical display lines. Interlacing the video simply fills only 50 percent of those lines, thus increasing the perceived framerate, while reducing file size, because only have of those lines are filled. I’m not an expert, however, so if you want more information, check out this wikipedia page: