Short for Dots per inch.
It is a measure of image or print resolution.
It dictates how many 'dots' will be used to render an image or text, per inch. For instance, a standard computer screen has a resolution of 72dpi, so there isn't much point in producing images with dpi values higher than this. For instance, GIF and JPEG images meant for a web site will be fine at 72dpi.
On the other hand, if the image is intended to be printed, then you would set up the image to have the appropriate dpi level. For instance a medium-quality image on a piece of A4 can be obtained with 125 dpi. A photographic-quality image may need to be 300 dpi or even higher. Of course, the printer must be capable of rendering at that resolution in the first place. For example a black and white laser printer can easily print at 600 to 1800 dpi. The print driver may allow you to select 'draft', 'quality', 'high quality' document modes.
You will also see dpi quoted for optical scanners that run to well over 1200dpi. These high values are used to scan in items such as photographic slides and film or large format documents.
As dpi goes up, then the image file size goes up much faster (double the dpi, you quadruple the file size). So it is a balance that depends on what you intend to do next.
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Click on this link: DPI