DPI is an acronym for Dots Per Inch.
Whether printed or viewed on some form of a screen, all computer output is printed as a matrix of nothing more than dots. On a black and white printer, there are only black dots. On a color printer or display, the spectrum might be represented by unique combinations of three colors – red, green, and blue, typically – that combine into a single visible dot of color.
Dots per inch is a measure of how closely those dots are placed to each other – in other words, how many there would be in one inch of display media.
The number of dots per inch a device is capable of supporting defines, among other things, its sharpness and clarity at presenting what is displayed. A higher DPI means more dots are used to display a specific area, and thus more dots can be used to represent fine visual characteristics.
Typically items are rendered “best” (a completely subjective term) when the computer presents information at the native DPI resolution of the device being used. However, it’s not at all uncommon to treat a device as having greater, or fewer, dots per inch to adjust the resulting size of what is being rendered.