An example of a Quickscope.
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 this, you must quickly tap your enter scope button(typically left click), and immediately after, shoot. Quickscopes look more professional and satisfying if the scope-in period lasts for a few milliseconds.
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.
Here are some videos showcasing quickscopes and drag scopes: