The higher the dots per inch the more detail can be crammed into a fixed width of physical space. A tighter grided network that can register more detail, or show more resolution.
Printers use DPI to express how many dots they can squeeze into a square inch of paper. A low DPI might mean the text looks too blocky, whereas a high DPI (600+) is usually a nice readable print and it's not too pixelated.
When moving a mouse across your mousepad often the computer isn't being sent the very small movements you make. This is - in part - due to the low DPI of most mice. You need to move the device a millimetre or two for it to register as being moved (another part of this is because we're still using a mouse with a ball in it and - as a heavy object - it's liable to remain still AND NOT ROTATE after only a small nudge. A laser, like the MS Intellimouse, is probably the best solution to mouse DPI woes).
The DPI conspiracy