A Nuclear Bomb (sometimes "Atomic" or "Atom" Bomb) is any nuclear device (where "Nuclear" indicates "deriving power from nuclear chain reactions") designed to be dropped from a height over a target, utilizing gravity as the delivery mechanism. Nuclear bombs were a large part of the early Soviet and US nuclear deterrent in the Cold War, and as such, captured the imagination of people across the world as instruments of terrible power. They have appeared in all kinds of fiction from the 1940's onwards, and have made their way into video games for obvious reasons.
In Real Life
In real life, Nuclear Bombs make up a tiny portion of modern nuclear arsenals - the majority of nuclear devices are fitted to ballistic missiles. The destructive radius of a device varies wildly - and is subject to the "inverse square law". In layman's terms, this means that the destructive radius of a device squares when the power (yield) of a device is cubed. This means that the more powerful a bomb is, the less destructive it is proportional to its yield (the destruction from a 1 megaton device then is only 10 times more powerful than a 10 kiloton bomb, despite having 100 times the yield). This then means that most bombs are actually of a lower power, because they are more efficient this way.
There are two kinds of nuclear weapons - Fission and Fusion (so called "Dirty bombs" are not nuclear weapons, they are radiological weapons, having more in common with poisons than with nuclear devices). A fission bomb initiates a nuclear chain reaction in one of two methods - Implosion style (where a core of a fissionable material, Uranium - 235 for example, is surrounded by explosive charges which detonate inwards towards it, causing the "Core" to collapse in on itself) and Gun style (2/3 of the material is on one end of a "barrel", 1/3 at the other. The 1/3 is fired at the 2/3 with explosives). Fusion bombs (aka Thermonuclear devices) contain materials such as Tritium or Deuterium (both isotopes of hydrogen), which are fused together to create heavier materials. A large amount of energy is released when this occurs, however an enormous amount of heat is required for the fusion reaction to occur - this is provided in all current fusion devices by containing a small fission device inside the casing. This is a "multi-stage bomb". Bombs that do not use an initial fission stage do not currently exist, and are known as "Pure Fusion Bombs".
Nuclear weapons have been used in anger twice in history - the 6th and 9th of August, 1945, at Hiroshima and Nagasaki, respectively. The two bombs resulted in 220,000 deaths, most of these from the firestorms generated, not the actual explosions themselves. After the second world war, the United States was the only country in possession of Nuclear Bombs until 1949, when the Soviet Union tested "Joe-1", the first non-American nuclear device ever built. This led to the escalation of the Cold War.
Today, there are estimated to be 27,000 Nuclear Weapons in the possession of at least eight countries, 96% of them between the United States of America and Russia.
In Video Games
Naturally, nuclear weapons in most video games are often nothing like their real-life counterparts. They usually come in two different flavors - grossly underpowered and grossly overpowered. But with more powerful graphics engines and physics modelling, along with better hardware, we are starting to see some games which walk a line a bit closer to reality. With a game such as Crysis able to provide some very realistic blast effects and destruction modelling, in relation to the type of weapon used by the player.
In Call of Duty 4: Modern Warfare, one can participate in the player character's feeble, shambling steps during his death throes; whilst witnessing the aftermath of a small nuclear detonation in a city. It may be one of the most realistic depictions of a nuclear weapon in a video game so far.
In addition, videogame nuclear devices will always have a mushroom cloud, regardless of it's yield, and radiation will sometimes be completely ignored. When it is not ignored, it is never treated in a realistic manor. The discrepencies in real life and videogame nuclear bombs can be the result of the creators not doing the research, or deliberate design decisions (videogames are subject to the "Rule of Cool" - it doesn't matter how unrealistic something is, if it's cool, you can put it in).
Grossly underpowered Nuclear Bombs are usually featured in real-time strategy games and shooters. Examples include the "Fat Man" launcher from Fallout 3, the Redeemer from Unreal Tournament, and the Nuclear missile from Command & Conquer: Generals. This is always the result of gameplay balancing.
Grossly overpowered bombs are not usually a part of gameplay - but rather a plot element. Examples include the nuclear device detonated that destroyed almost the entire of Racoon City at the conclusion to Resident Evil 3. This is highly implausible, as even a medium sized city requires multiple devices to completely destroy. The effects of Nuclear weapons are usually played up to add tension to a thriller plot, or as in the case of RE3, to provide closure to the story.