Quickscopes are, as the name suggests, killing someone while in scope of a sniper rifle within a very short period, such as 0.3 seconds.
To execute Quickscope, you must quickly tap your enter scope button (default: RMB), and immediately after, shoot. Quickscopes look more professional and satisfying if the scope-in period lasts for a few milliseconds.
How this works
Most Sniper Rifles have 100% accuracy when zoomed in, meaning if players just open the scope then fire, the bullet will go straight to the center of the screen. Quickscope relies on the fact that players have caught sight of their enemies (or expecting them), then just before facing them, using Quickscope will guarantee a center bullet, which hit the enemies as they appear. This is best done in Free For All (Sniper) due to players having no allies at all so they won't need to worry about wasting shots on teammates.
Advantages and Disadvantages
Quickscoping is a very useful skill to have at close range and long range too. At close range, it is useful to quickscope because many players run around hastily, therefore it is tremendously difficult to keep following them with your crosshair. Also quite useful at long range, because a quickscope bullet will "reach" the other player quicker because of the shorter scope in time.
Disadvantages include requiring the player to become immobile, as a quickscope(and most other sniper shots) becomes increasingly inaccurate when moving. Missing a quickscope shot can have grave consequences, especially when immobile. Quickscopes also require a getting used to where the sniper rifle crosshair is aimed out of scope(exact center of the screen), so it is recommended for experts mostly.
Only two sniper rifles are not able to do quickscope: the MSR and Barrett M98B. It's because these guns have a style of scope which take 1 second to display, making these guns not popular among the quickscoping community.