1080i refers to an interlaced HDTV signal with 1080 horizontal lines and an Aspect ratio of 16:9 (1.78:1). All major HDTV broadcasting standards include a 1080i format which has a Resolution of 1920x1080, however there are other formats, including HDV and AVCHD for camcorders, which utilize 1080i images with a resolution of 1440x1080. Confusing matters even further, a 4:3 (1.33:1) image broadcast in 1080i (1920x1080) uses the center 1440 Pixels for the image, with 240 pixel wide pillarbox borders on each side.
For HDTVs themselves 1080i can also have different meanings. While you might expect a 1080i HDTV to have a resolution of 1920x1080, this isn't always the case. For many flat panel 1080i HDTVs the actual Native Resolution is 1366x768. While these TVs accept 1080i input signals (as do all HDTVs) they must scale them down to their native resolution, resulting in somewhat lower picture quality.
The obvious advantage to 1080i resolution is that it uses the highest (standard) resolution available for TV signals and consumer electronics equipment. It can also be better for older (analog) interlaced video, which was a standard format to use for TV broadcasting (and video tape) for decades.
Although there are HDTV broadcasts in 1080i, many channels broadcast in 720p. These broadcasts must be upscaled by any 1080i HDTV. This is particularly true of sports channels. Additionally, regardless of the capabilities of the display, the video encoding currently used for HD video allow a maximum of 30fps for 1080i, while 720p may be encoded at 60fps.