Radiation is a term that covers a broad range of objects from harmless radio waves to deadly and destructive cosmic rays. To radiate means to emit, and in physics terms, Radiation refers to energy in the form of waves or moving subatomic particles emitted by an atom or other body, as it changes from a higher energy state to a lower energy state. But none of you need to know that. Radiation is divided up into two types - Ionizing radiation and Non-Ionizing radiation.
Radiation, or radiate, literally means something that moves outwards from a central point to create a radius.
In Video Games
Generally speaking, in a video game (or all fiction, for that matter), if somebody says the word radiation, then something bad is probably about to happen. Although in reality, radiation exists everywhere in the universe (and indeed, light is a kind of radiation), the only time radiation will ever be mentioned in a game is if it is detrimental to your own health. Radiation may be used in a variety of ways as a gameplay mechanic. In Call of Duty 4, for instance, in the Chernobyl level, if you stray out of the bounds of the level, you are warned against "hazardous levels of radiation", which will kill you if you ignore the warnings (this is largely nonsense - the levels of radiation surrounding Chernobyl in the 1990s, when the game was set, are virtually non-existent at any great distance from the core of the reactor itself. The mechanic replaced the typical "land mines" seen in previous Call of Duty games).
In Real Time Strategy games, radiation will often be represented in the aftermath of a Nuclear Detonation as an environmental hazard similar in nature and functioning to typical "poison" behavior in said games. Command & Conquer: Generals, for example, has radiation that is represented by orange clouds, and should an infantryman wander inside, he will start taking damage. Radiation is released in the aftermath of Nuclear Reactors being destroyed, certain neutral structures being destroyed, or a Nuclear Missile being detonated over a target area.
In most games the type of radiation is likely to be gamma radiation, the type of radiation that is left when a nuclear bomb detonates. Since it is the most powerful (has the most energy within each proton) it will hurt players/factions/characters due to the affects of ionization.
Ionization is when an atom absorbs the radiation and then the radiation will cause the atom to form an ion (an atom with a positive or negative charge), if the atom is a part of a cell, it will ruin the atomic structure of said cell and cause the cell to create different chemicals, defect and then lead to rapid growth of a cell or cell group. The defects will grow into a tumor.