Since we are considering mating threats, this is a good moment to consider one additional type of pin: a pin of a piece not to its king but to a mating square. A mating attack frequently depends on whether you can get an attacking piece onto some square in particular—e.g., whether you can land your queen next to the enemy king. If one of your opponent's pieces blocks a move like that, it's pinned: the game is lost if it moves. The pin may not be as strong as the usual pin to the king, since a piece pinned to a mating square at least can move legally; if it has a strong threat—a check of its own to deliver, for example—it may be able to get away with leaving its square. It still is important to see pins of this kind, because the pieces subject to them often are prey to capture and useless as defenders of other pieces.
Notice in this first diagram that Black’s king is trapped on the back rank and has no defenders there. This suggests a mating possibility if White can get his rook on e1 down to e8. Alas, he can't do it because Black’s bishop is in the way. But therein lies a tactical idea: Black’s bishop is pinned to a mating square (e8); if attacked it cannot afford to leave the e-file. So White throws a pawn at the bishop with f2-f3 and takes it a move later.
The key to the position lies in examining Black’s king and seeing its vulnerability. From there you reason backwards to ways you might take advantage and then to obstacles that prevent you from doing so; finally you consider whether the obstacles are themselves vulnerable by virtue of their defensive role.