The computer, or as the French say, l'ordinateur, is a device that computes. Especially, say, a programmable electronic machine that performs high-speed mathematical or logical operations or that assembles, stores, correlates, or otherwise processes information. Comes in a variety of colors. Why, you're using one right now. No, seriously, you are, I'm not kidding. Look down.
In our more narrow definition here at Encyclopedia Gamia, the definition of a computer can be narrowed down to a Personal Computer. In essence all video game consoles are computers, each capable of logical operations, but a Personal Computer differs because it offers a full Keyboard/Mouse setup, in addition to a few other customization aspects that most consoles do not offer.
Similarly for arcade games, arcade system boards are also technically computers, and were precursors to the upgradable nature of modern PC systems, but they differ in the sense that they are dedicated game machines, whereas personal computers are general-purpose machines that can be used for practical as well as entertainment purposes.