Simply put, a device capable of n DPI means that this device can produce, on paper or on a screen, dots small enough to fit n of them in an inch (2.54 cm). This can be different from the actual image resolution, since often a pixel on the image is composed of several dots.
Computer monitorss have a resolution of about 72 to 110 dpi. That is, they have 72 pixels per inch, and an image 140 pixels across will occupy about two inches. Television resolution is at most half of that, and usually much lower.
Graphic printers must have a much greater resolution, because the human eye is much better at making out small points on paper. Text can be rendered with good quality at 300 or 600 dpi. Color images need even higher dpi, with professional equipment leveling around 4000 dpi.
The high dpi needed for a printed image poses great problems to digital photographers and artists, who must acquire images in a very high resolution (thousands of pixels), before being able to print them with an acceptable quality. The rapid increase in the megapixel count on digital cameras is driven primarily from this need.