A Sacrifice is a bold move where you give up a piece to gain a decisive advantage. It is the ultimate proof that the activity and position of your pieces can be worth more than their points.
In a sacrifice, you aren't "losing" a piece—you are "investing" it to open lines or trap the enemy King.
One of the most famous sacrifices is the Greek Gift. You sacrifice your Bishop on h7 to lure the Black King out of safety. While it looks like you just lost a Bishop, the move actually creates a "hole" in the defense. Because the King is now on h7, your Knight and Queen can jump in with checks that the King cannot escape.
Crucially, a sacrifice forces your opponent into a "take it or leave it" situation. If they take the piece, they fall into your trap. If they don't take it, they usually lose an important pawn or square anyway. This is why calculation is so important—you must ensure the follow-up is unstoppable.
In the first task, notice the Smothered Mate. By sacrificing the Queen, the Black Rook is forced to move to g8, which physically blocks the King's only exit. This allows the Knight to deliver a mate that wouldn't be possible if the exit was clear.