Quickscoping is older than the Call of Duty series, but it's a bigger problem in that game than elsewhere.
The literal definition of quickscoping is exactly what you'd think - aiming down the sights of a sniper rifle for a short period before firing. This is in contrast to "hard scoping" where you spend most of your time aimed down the sights and lining up a shot before firing.
Sniper rifles generally trade situational awareness for accuracy over long range (ie, I'm staring down this scope and lose my peripheral vision, but I can take a headshot from across the map). Quickscoping is a method of maintaining both, by only looking down the sights and quickly aiming when a target is available. In that way, it could be viewed as a mark of skill, a risky endeavor, or a potentially cheap strategy.
However, in CoD parlance, especially on consoles, there is a more insidious use for this technique. The game features an aim assist that will tend to pull your crosshairs towards an enemy as you start to aim. This is intended to offset the inaccuracy of using thumbsticks for aiming.
However, when using a sniper rifle, it means if you roughly line up a shot, then aim down the scope and quickly fire, there is an increased chance that you will get a hit/headshot without having to really "aim" for one. Once you get the timing right, it's incredibly easy to pull off, and thus is viewed as more of an exploit than a legitimate strategy.
Especially in CoD: Modern Warfare 2, it wasn't unusual to find folks running around the map at top speed in a ghillie suit while carrying a massive sniper rifle, and killing you in close quarters despite the fact that the balance of the game should favor other weapons in this kind of engagement.
This isn't nearly as big a problem in TF2 because there isn't aim assist (or at least, not nearly as strong as console CoD titles) and also because all of the sniper weapons require a charge time before firing at full damage. It's still possible to "quick scope" with the meaning of "spending little time aiming before firing" but the tactical advantages of doing so are far less.