Dithering is the process of juxtaposing pixels of two colors to create the illusion that a third color is present. It is basically digital stippling because it is the creation of a pattern simulating varying degrees of solidity or shading by using small dots -- in our purpose, pixels. When using a darker hue, the denser the dots, the darker the apparent shade; when using a lighter shade, the denser the dots, the light the apparent shade.
Since in most cases colors in a palette are limited, artists make use of dithering to maximize the use of colors. It is used to diminish the harsh transition from one color to another therefore reduces the overall sharpness of an image, and it often introduces a noticeable grainy pattern in the image. It is also used to create an illusion of depth.
Personally, I just use detailed dithering on static artworks such as posters and backgrounds, but there are some incredible ways to apply dithering in animation as well. When applying shading to an item with a limited color palette, there is such thing called ordered dithering (example below) with different patterns that the artist can use.