What is "quickscoping"?
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.
Quick scope is the act of zooming in or "aiming down the sights" with a sniper rifle and firing almost immediately after.
The term became really famous/infamous with the Call of Duty series, but the action itself goes way back to early FPS like CounterStrike. Sniper rifles do full damage, but were terribly inaccurate unless scoped. As soon as you hit the "aim" button the accuracy immediately became perfect, so staying in scope (known as "hardscoping") wasn't needed.
games like TF2 attempt to nerf quick scoping by making sniper rifles weaker unless scoped (which doesn't really work as it can still one-hit many classes on headshots when it's not "charged"). others make it so there is a "ready time" where the gun is still pretty inaccurate while first scoped.
Quickscoping is exactly what you just said. Being ready to take a shot, zooming in, then immediately firing. However, your bit about TF2 is wrong.
A sniper rifle (stock) can do between 50-150 damage without hitting someone's head, usually. It does such a wide range of damage because it charges while you are scoped in, if you look to the right side of the screen while scoped you'll see an indicator. When this indicator is full you will get maximum shot damage, be it headshot or body shot. The damage for a headshot is also similarily spread out between 150-450, based on charge percentage. You can kill a fully overhealed heavy with one headshot if you have a fully charged sniper rifle, for example. You can also kill a scout with a fully charged bodyshot. A good sniper in TF2 can take full advantage of quickscoping and put out a larger spread of damage than a single fully-charged shot will.
Imagine a medic, an engie, and a pyro all coming at you from far enough away. If you are able to zoom in and click their heads, effectively killing all of them but the pyro in one shot each, you've done better than you would have if you waited the couple of seconds it takes to fully charge then waste a huge damage headshot on someone who doesn't even have that much health. Using a fully charged body shot is similarily ineffective if you can actually hit their heads anyway. There is absolutely a reason to quickscope in TF2.