Pixels per inch (PPI) is strictly concerned with the output resolution, and the value attached to a particular image file is only a suggestion.

The image file itself is composed of a set number of discrete pixels, which are to a picture what atoms are to molecules. Unless and until you change the number of pixels, the image quality when the image is displayed at the same size on the same output medium will remain the same. If you tell your printer to print a given image file at a certain fixed size (say 8x10 inches, for the sake of argument), it doesn't matter in the slightest whether the resolution tag in the image file says it's a 72PPI image or a 300PPI image; if the file is otherwise identical (that is, if it contains the same pixels in the same colour space), the prints will be identical. In the same way, a 1000 by 1500 pixel image will display at the same size on a web page (barring CSS or HTML scaling, which can stretch or squish the image, usually with a significant loss in quality).

Yes, you do generally want higher PPI values for output when you can get them (up to a limit -- if the resolution is higher than the output device can support, you'll actually lose quality), but that means printing larger image files at smaller physical sizes, or displaying the image on a screen with more tightly-packed pixels (which amounts to the same thing). You may need to calculate the appropriate number of pixels for output, which does involve PPI, but that is usually to match the output device's resolution, the desired print size, and the image file.

(If you want to print to a dye-sub or "light-jet" printer, you'll get the best quality if the pixels in your image match the printer's pixels. If the printer prints at 400PPI, and you want your image to print at 16"x20", it ought to be a 6400x8000 pixel image, otherwise the printer will have to scale it for you, and that doesn't always go well.)

The PPI value embedded in the file is used mostly in pre-print processing applications to determine the image size. That includes things like desktop publishing apps (e.g. Adobe InDesign), third-party raster image processors (RIPs), and multi-up gang printing applications that have to figure out how best to minimize paper waste. 300PPI is pretty much industry standard for prepress work, and image sizing is usually done in an image editing application to minimize scaling artifacts and optimize sharpening, etc.