A video game is these days mostly synonymous with console game, but the term is sometimes used to also describe computer games (even when this sounds wrong). The term "video game" usually refers to a consumer product, that is, a game system that a player buys to his/her personal use; it is usually not used of arcade games or other such games that are not designed for consumer home market. However, like in case of computer games, the term is these days sometimes applied to arcade games as well. So, these days, the term is often used to span proper "video games", console games, arcade games and computer games - everything that draws on a screen of some sort, or a reasonable facsimile of thereof.
In historical meaning, a video game system is a game system that uses video output (usually to television or a monitor) as the display method. The earliest video games marketed to consumers were units that were connected to TV and had a fixed set of games. Later the game consoles came to market; these were usually based on removable ROM cartridges. Modern game consoles also use CD-ROM and other laser disc technologies (most recent consoles use variations of DVD format). The game consoles were a great leap forward from video game systems; The oldest video games were not necessarily even digital and expandability was a far-away dream... In general, the growing trend in video games is that they're becoming less and less dependant on specific video game systems (portability), and games are themselves becoming modular and expandable (reuse of game engines in other games, expandability and "moddability" of games).
As is seen, the difference between "video game" and "video game system" is difficult; Historically, "video game" referred to both the system and the game(s) the system ran (since the system was capable of running only certain built-in games), and nowadays, "video game" refers to a single game played on a "video game system".
People still use the term "video game" to describe "a game that you play on the TV", so the term persists.
Games themselves often involve one or more players (in case of computer games and some console games, also other players over a network or other communications channel), and often a game AI. The games are controlled via various types of controllers (joysticks, joypads, trackballs, light guns, mice, and keyboards, just to list the most influential ones). As per the definition, things are displayed on screen in graphical or alphanumeric form (some games don't need graphics to survive). Many games also include sound effects and music.
See also: Video Games (E2's gigantic game list) | 
