Removing the Defender
The idea behind removing the defender is simply to get rid of the support a piece has. You remember that one of the five ways to meet a threat to capture is to defend the threatened piece. Well, this tactic gets rid of that defender.
1. e4 e5 2. Nf3 Nc6 3. Bb5 Nf6 4. d3 a6
The e5-pawn is under attack by the knight on f3. It is also defended by the knight on c6. So White removes the defender and picks up the pawn for free one move later.
5. Bxc6 dxc6 6. Nxe5.
Here's one a bit more complicated.
1. e4 e5 2. Nf3 Nc6 3. Bb5 d6 4. d4 Bd7 5. Nc3 Nge7 6. Bc4 exd4 7. Nxd4 g6 8. Bg5 Bg7.
Black threatens to capture the knight on d4. He has a converging attack with his knight on c6 and bishop on g7, while White only defends with his queen.
White did not defend the d4-knight. What does that mean? Black decided to find out the hard way.
White's knight went to d5 so it could get to f6 with a check. But f6 is defended—by the bishop on d4. Therefore, White removes it.
10. Qxd4! Nxd4.
So what did White get for his queen? How about a checkmate?
11. Nf6+ Kf8 12. Bh6 checkmate.
All these tactics, whether long-range or short-range, are possible with the right buildup. It is merely necessary to know what you are looking for and then find a way to implement it.
The hard part is that you have an opponent who doesn't want you to pull off any of those tricks. So you need to learn to look at whatever position is in front of you with a keen eye for any of the patterns you have learned. When one begins to take shape, play for it. The rewards will come soon enough.