1080i refers to a video stream which is 1080 lines (the maximum resolution of HDTV) and interlaced. A typical interlaced video delivers each video frame as two fields. The first field will be the 540 odd scanlines of the frame, and the second field will be the 540 even scanlines. Because the fields are delivered (and usually recorded) at slightly different times, if there is motion in the frame it is not sufficient to simply mesh them together to produce a complete frame on a progressive display. To accomplish this, deinterlacing is required.
The majority of broadcast HDTV comes as 1080i. Notably, in the USA, all TV networks except Fox and ABC use 1080i, as do most cable and satellite HD channels. Fox and ABC use 720p. Displaying 1080i content and deinterlacing it is the most hardware intensive type of video playback. It tends to require video decoding hardware, or a computer that is in the range of a 3ghz Pentium 4 or similar. (Such a computer may still have problems if the stream is encoded in a format that takes high CPU to decompress, such as some forms of h264.) It is not a problem for modern dual-core processors.
1080p video, which does not use interlacing, is not currently broadcast, but will be found on some Blu-Ray DVDs and downloaded video sources. Almost all computer monitors, and many HDTVs, are driven with a progressive video signal and deinterlacing is required to play 1080i.