Most commonly associated with shooters and competitive multiplayer games, an instakill is any move, attack, vehicle, or weapon that will instantly kill a player who is at full health. Also seen in roleplaying games, usually as some variant of a "death hex"- a low-frequency spell that will instantly destroy an enemy monster. Stealth games (and action games with stealth elements) also feature these heavily.
The most common instakill in shooters is with a headshot- a sniper headshot almost always is a one-hit kill, although in more realistic/hardcore games, a headshot with ANY weapon will usually kill instantly. The skill to land a 1-hit kill is usually lauded- Halo gives out medals for sniper headshots, and Modern Warfare 2 gives players a "One Shot, One Kill" bonus for taking their enemy out with a single shot.
The other main one-hit kill in shooters is a combat knife, given to players in addition to their gun in most modern games, although in older titles it was often a selectable weapon you would have to trade a gun for. Knives are almost always instant kills in games today, often associated with a lunge that will move a player towards his target a little bit, extending the effective range of the knife quite a bit, although it is still limited to close-quarters encounters.
Certain other weapons such as rocket launchers and shotguns can one-hit kill in the correct circumstances. Weapons with the ability to kill in one hit are usually restricted as "power weapons"- there is usually only one or two on any given map in an arena style shooter, while they will be (usually) restricted in some way in class and loadout-based shooters.
There are also certain game modes in some shooters that are based around one-hit kills (known as Instagib in some contexts). Quake II's Railgun and Unreal Tournament's Instagib Rifle are examples.
Instant kills are a staple of turn-based RPGs - both for party members and for certain enemies. Usually the mage of the party can learn a spell that has a chance to kill its target instantly; in the Dragon Quest series, for example, this spell is called simply "Death," and at higher levels it can be cast on multiple enemies. Overpowered though it may sound, such a spell will always cost a lot of magic points, has a low hit rate, and will usually not work at all on higher-level enemies or bosses. Such limitations lead to death spells usually being regarded as ineffective by most players. There are exceptions, such as enemies that are weak to instant kill spells, but to prevent unbalance this ability is highly restricted in most RPGs, at least for the player. Some bosses can instantly kill party members or even the entire party, either by casting an instant kill spell of their own or simply being too powerful for the player at the current stage. The latter scenario can indicate that the boss is unbeatable.
The instant kill is a staple of the stealth action genre- the bread and butter of these games consists of taking enemy guards out quickly and quietly. The traditional shooter technique of shooting for the head still applies, but steal action places many more tools in the hands of the player- tranq guns will usually neutralize an enemy in one strike, and if the player sneaks up on an unsuspecting guard snapping their neck or putting them in a choke hold can also quickly eliminate the problem.
One of the most dangerous instant killers is the game environment, which can kill players in any number of ways. A fall from a sufficient height will kill in most games, as will jumping out of a multiplayer arena that is floating, such as Facing Worlds in Unreal Tournament, or Lockout in Halo 2. Lethal hazards can also be found in the level itself, such as the bullet train in Halo 2's Terminal, which speeds through the middle of the level at random intervals and kills anyone on the tracks.