A "quick scope" is a kill in first-person shooters, where the fast aiming with a sniper rifle and the direct pulling of the trigger allows you to use the high damage of snipers without suffering from the disadvantages of limited vision and a long aiming process. Thus, even in 1 vs 1 duels at short range, considerable killstreaks can be achieved with the sniper rifle.
In a quick-scope, you shoot directly after switching to the scope. To do this, the weapon is centered as best as possible throughout the game and a quick combination of aiming and pulling the trigger is executed as soon as an enemy is in the center of the screen. The opposite would be the so-called "hardscoping", where you aim through the scope the whole time.
In many games the damage of sniper rifles increases massively when you aim with them, hits to the upper body are often enough for a frag. In combination with aim assist, quickscoping on consoles becomes a comparatively simple affair with a bit of practice, which can be annoying for the opponents relatively quickly depending on the map and game.
The first entry on the topic of quickscoping appeared in the Urban Dictionary back in 2007, and at the beginning it mainly referred to Counter Strike. However, the topic really took off with the increasing popularity of the Call Of Duty series, where it was comparatively easy and massively abused.
As old as quickscoping is the discussion about whether it is skill or exploiting game bugs. Depending on the opinion, quickscoping is true art, a risky and inefficient endeavor, or a cheap mechanism where you simply exploit bugs in the game and benefit greatly from the aim assist.