When dealing with an image, there are two "pixel density" issues you are dealing with. First is matching the resolution of the image to display on a monitor, and the second is giving it a resolution that works with your printer.
A monitor usually displays at 72 or 96 DPI, although "inch" is more of a reference than a hard value because at either scale, the size and resolution of your monitor will determine how big the inch actually is. You don't want to have to scale your image when it is viewed on a monitor or it could easily become blurry, distorted, or pixelated. To ensure your image is viewed at its best, usually being between 72 and 96 DPI is best.
Most image viewing programs will automatically scale a picture that has a higher resolution so that it fits on the computer screen, but when you don't have control of what image software the viewer will be using, you want to just set it to a resolution that will work for all displays. Web clients, for example, don't pay attention to dimensions, they only pay attention to pixels, and if you have 1000 pixels across, an image with 500 pixels of width will take up half of it. So you set it to a resolution that is based on a certain DPI and base the width on that. Most image editing programs allow you to work in raw pixel size or in dimensions and DPI scale. They are effectively the same thing, DPI just gives you an easier number to remember. In the case of a monitor, PPI and DPI are one in the same really, or at least could be said to have a 1:1 ratio.
A printer is very different, however. A printer that can print at 1200 dots of ink in one inch is referred to as "1200 DPI". But usually the images printed are more like 300 or 400 PPI. A 72 PPI picture that is 720 pixels tall and 576 pixels wide is technically 8" x 10". But if you sent the raw image data to a 1200 dpi printer, it would end up about a half inch wide and just a little taller than a half inch. So the PPI is scaled when you send it to your printer, either by the image software or in the printer driver.
But a 72 PPI image that large would not look good on a printer if you force it to be that size. You don't need it to be 1200 PPI to match the printer's 1200 DPI, but an image at 300 or 400 PPI would probably be as perfect to the naked eye as a 1200 PPI image. So you set it to 400 PPI and the printer prints 9 dots for each pixel (3x3 dots make up each pixel on the image). Keep in mind, the image software or the printer's driver will usually scale it to whatever the image should be for size, so technically the scaling from PPI to DPI is done without you messing with it, but when converting images, you might at least want to be scaling the image's pixel density to match the device you are intending to display it on.
In practice, I usually keep all images at 300 PPI (regardless of format) when working on them, and then scale down if I am going to display the image on a monitor as a final result. That way if I go to print it, I have a high enough resolution that it will come out good.