Dithering was most widely used by older 2D games. The natural hardware blending/blooming effect of older CRT displays and the color bleeding of composite cables allowed developers to use dithering to simulate more colors or transparency effects than was possible in the system hardware.
This technique was often used by games on the Sega Mega Drive/Genesis console, in order to simulate the higher color depth of arcade systems (including Sega's own arcade systems) and/or to compete with the SNES's higher color depth.
Sega Mega Drive/Genesis versions of mutliplatform games often used dithering in places where the arcade or SNES versions don't.
Dithering was also used for textures in early 3D games, such as Tomb Raider and the PlayStation port of Ridge Racer.
Sega Saturn versions of multiplatform games often used dithering to simulate fog and transparent water, which wouldn't have been dithered in the PlayStation or PC versions.
The PlayStation used full-screen dithering, to make the image look smoother on CRT displays with composite cables. PlayStation games frequently used dithering to simulate a higher color depth, or to prevent color banding. Many PlayStation games thus have a grainy look.
Dithering is still used to an extent today in some 3D games, usually in conjunction with deferred shading, or to prevent color banding.