Pixels Per Inch, or PPI, is a measure of density or resolution (resolution in the sense of fineness of detail, which again refers to the density of information, rather than dimensional resolution, or the physical parameters). Not all devices that can be used to view photos have the same pixel density. Computer screens tend to range from 72ppi to 109ppi in terms of screen pixel density. Relatively speaking, this is fairly low. Newer hand-held devices have screen densities anywhere from 135ppi to as high as 360ppi. Such devices would be the likes of smart phones, tablets like the iPad, Nexis Android tablets, and forthcoming Windows 8 tablets. Printers can range from as little as 100ppi to as high as 720ppi for ink jet printers.
The value of knowing PPI, or pixel density, is in knowing how large to export your processed photos for viewing on various devices. If you intend to produce images for viewing on a standard computer screen, you can generally assume around 72-100ppi, and exporing at 4x6 to 5x7 size from a physical screen space standpoint will usually produce photos that are easiest for potential online viewers to observe. If you intend to produce images for viewing on portable hand-held devices, you should target a higher resolution, around 150ppi would be better.
When it comes to print, knowing what PPI to print at can be extremely important, as it can greatly affect how large you can print. If you intend to print your photos at 8x10, you are probably interested in printing at least 300ppi or higher...even as high as 600ppi (Canon and HP printers) or 720ppi (Epson printers). Most photos, excepting some forms such as architectural photography (and even in some senses architectural photography) contain more detail than can be resolved in a 300ppi print at 8x10 for normal viewing distances, so printing at higher resolutions is often necessary to fully realize the detail you may have in a photograph. When printing at "native" print size...or the a size relatively close to the native image dimensions of your photographs...which tends to fall in the range of 11x16" to 17x22", you will probably want to print around 300ppi. At normal viewing distances, this should capture all the fine detail in your photo well enough to be observed by most viewers, and will usually require little to no scaling of your image to print (meaning you'll get the best results at the largest native size.) Finally, if you intend to enlarge your photos, you will probably want to print at a lower resolution. Average enlargements, which might be up to around 24x36" or so, can often be done at around 150ppi, however 200ppi tends to generate slightly better results for average viewing distances in normal papers, and in many cases even canvas. For significant enlargements, up to 40x60, you will want to print at 100ppi to 150ppi, with the expectation that such photos will be viewed at a distance of at least several feet if not more.
Understanding pixel density, or PPI, and how it relates to the physical dimensions of the images as observed...in either print or on screen, is important to producing the best quality "final output." Exporting or printing an image at an improper pixel density will produce less than ideal results, often softening detail that your viewers could otherwise readily observe.