1080p is the shorthand name for a category of HDTV video modes. The number 1080 represents 1,080 lines of vertical resolution (1,080 horizontal scan lines), while the letter p stands for progressive scan (meaning the image is not interlaced). 1080p can be referred to as full HD or full high definition although 1080i is also "Full HD" (1920x1080 pixels). The term usually assumes a widescreen aspect ratio of 16:9, implying a horizontal resolution of 1920 pixels. This creates a frame resolution of 1920×1080, or 2,073,600 pixels in total.
1080i is the shorthand name for a format of high-definition video modes. 1080 denotes the number of horizontal scan lines - also known as vertical resolution - and the letter i stands for interlaced. In the alternate format of high-definition video mode, known as 1080p, the p would stand for progressive scan.
1080i is a high-definition television (HDTV) video mode. The term usually assumes a widescreen aspect ratio of 16:9, implying a horizontal resolution of 1920 pixels and a frame resolution of 1920 × 1080 or about 2.07 million pixels.
In a tube-based television, otherwise known as a CRT, 1080i sources get "painted" on the screen sequentially: the odd-numbered lines of resolution appear on your screen first, followed by the even-numbered lines--all within 1/30 of a second. Progressive-scan formats such as 480p, 720p, and 1080p convey all of the lines of resolution sequentially in a single pass, which makes for a smoother, cleaner image, especially with sports and other motion-intensive content. As opposed to tubes, microdisplays (DLP, LCoS, and LCD rear-projection) and other fixed-pixel TVs, including plasma and LCD flat-panel, are inherently progressive in nature, so when the incoming source is interlaced, as 1080i is, they convert it to progressive scan for display.
1080p is expensive compared to 1080i or 720p. Unless you are into video editing, or extreme graphic computer use, 1080p does hardly anything at all. Most of the broadcasters produce 1080i or 720p on cable, satellite, or broadcast. Blu Ray is the only DVD format that puts out a 1080p signal. The picture quality is no better than 1080i or 720p is the general consesnus.
As for gaming, Xbox 360 and the PlayStation 3 games are usually 720p native, though some titles are being offered in 1080p resolution (also, the 720p titles can be upscaled to 1080i or 1080p in the user settings of those consoles).