1080p refers to a progressive 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, but the progressive HDTV broadcast standards in place right now only allow a resolution of 1280x720 (720p). Currently the only applications using 1080p signals are Blu-ray and HD DVD.
Creating a progressive 1920x1080 signal is problematic because of how MPEG-2 and AVC profiles are structured. Rather than having separate framerate limits for different progressive resolutions they assume a resolution of 1280x720 for progressive video. In practice this has turned out to be less of an issue than it might seem. By limiting 1080p video to the same framerate as 1080i (30fps), the same hardware used to decode 1080i video at 30fps and 720p at 60fps is generally capable of 1080p at 30fps.
Although all 1080p displays can produce a progressive image from 1080i video or an upscaled 720p image, some older models can't actually accept a 1080p input signal. Since its technically outside standard HDTV specifications, neither can 720p or 1080i displays. Newer 1080p HDTVs are designed to accept these full resolution progressive signals, and in fact are generally capable of displaying 1080p video at framerates of 60fps or higher. However, practical bitrate and hardware limitations make this a purely theoretical possibility at this time.
Since almost all HDTVs are progressive in nature, even when they're designed for interlaced display, 1080p is a natural evolution of display technology. As HDTVs have gained in popularity, so has proper (progressive) treatment of progressive sources, primarily from film. Since HDTV hardware isn't tied to a particular framerate there's no reason that both film and progressive video of all types can't be encoded to remain faithful to the source video.
As noted previously, 1080p video falls outside the standard encoding profiles for both MPEG-2 and MPEG-4 AVC high definition encoding. This means hardware designed to closely match those profiles may not be programmed to handle 1080p correctly. In addition, since 720p and 1080i HDTVs aren't able to decode 1080p signals it's always necessary to provide an alternate signal, either 1080i or 720p, for the majority of HD displays.