With respect to video the numbers 1080 and 720 represent the vertical resolution in LINES per picture height (L/PH) or TV LINES per picture height (TVL/PH) for the two most common high-definition (HD) video formats. Ideally a 1080p display should provide a 50% increase in vertical detail over a 720p display and double the amount of overall spatial detail, all things being equal. (It is important to note that most movies are not framed in a 16:9 (1.77:1) aspect ratio therefore, along with other resolution limiting factors, the actual vertical resolution of the visible video image will often be less than the full vertical resolution of the active picture height of the display screen.)
When referring to 1080p or 720p, etc., the “p” is used to distinguish a progressively scanned (aka, non-interlaced) video image process from an interlaced video image process. Progressive scanning or progressive scan video is the process of displaying (or capturing) video images by scanning ALL of the horizontal scan lines, from top to bottom, within a single pass for each and every FRAME of video without subdividing each frame into two individual, lower bandwidth, sequential video FIELDS. Progressive scan video is consistently superior to interlaced video and should always be given preference if image quality is paramount.
Furthermore, for a given display size a 1920×1080 display will allow you to sit approximately 35% closer than a 1280×720 display. Conversely for a given optimum viewing distance you will be able to use a 1920×1080 display that is approximately 50% larger (diagonally) than a 1280×720 display. For example, with all the other factors being equal, at a nominal distance of 8 feet (96 inches) the optimum display sizes are approximately 41 inches for a 1280×720 display and 63 inches for a 1920×1080 display.
If a person wishes to achieve the most immersive viewing experience¹ the recommended field of view (FOV) between the viewer(s) and the active picture area of the display should be approximately between 30 and 40 degrees. The only HD display resolution capable of meeting such a requirement—without exposing most viewers to discernable picture structure when seated at the proper viewing distance—are displays that reproduce a full 1920×1080p or higher video image.
Virtually all high-definition digital video produced for home use, especially content that originates from film, is encoded, processed and mastered as 1080p/24 digital video for each of the two competing high-definition video optical disc formats. As a general rule of thumb for most—if not all—high-definition source material that is encoded as 1080p/24 the less video processing the better providing your display (and HD video source) can properly handle this HD video format. (The number 24 as written above refers to the video frame rate, aka the vertical refresh rate on a display, in frames per second, which must be specifically supported by the display in order for a viewer to benefit from this HD video format.)