A computer game is any sort of game that is played using a computer.
A computer game is not necessarily a video game, or vice versa; for instance a text-based role-playing game could be played verbally by a blind person, which is clearly no longer a "video" game, and the first generation of video games, such as Pong, used dedicated electronic circuitry not even remotely resembling a computer.
The usual distinction today is rather subtle; a game will be a "computer game" if it is played on a general-purpose computer, but a "video game" if it is played on a computer that is specialized for game playing. Computer games will typically feature a wider assortment of direct controls exploiting the full computer keyboard, while video games tend to use more layers of menus, or motion sequences (up-up-down-left, etc) via the game controller. The most important distinction between computer and video games arises from the fact that computers have high resolution monitors, optimized for viewing at close range by one person, while home video game consoles use a much lower-resolution commercial television as their output device, optimized for viewing at a greater distance. As a result, most computer games are intended for single-player or networked multi-player play, while many video games are designed for local multi-player play, with all players viewing the same TV set.
Formerly, video games tended to need and use less computing power than computer games, but with the increasing power of video game hardware, that distinction is nearly erased, and many games are now produced for both computers and video game systems. Video game manufacturers usually exercise tight control over the games that are made available on their systems, so unusual or special-interest games are more likely to only ever appear as games on general-purpose computers.