The pleasing solution to the position pictured here was published by Gioachimo Greco in 1619. Black’s king looks reasonably secure, but White has a queen and rook trained on h8 and a queen and bishop trained on f7 (with the bishop aimed through to f8), and Black has nothing but his king to protect either square. Your goal is to create an open diagonal and file with pressure down each. Your possible tools for the purpose: captures of pawns; threats and captures on squares they protect; and mating threats against the squares they occupy. In this case captures work best. You start by removing the f7 pawn with Qxf7+, forcing Black to move his king to h8. Now what? White has created diagonal pressure against g8 (indeed, he has redundant pressure on the point, which is important; it means he can afford to move his queen). Now he wants to add pressure down the h-file, so he needs to open it. He plays the sacrifice Rxh7+. Black’s forced reply is KxR—and now White mates with Qh5. It's another case where the queen is able to zig-zag helpfully between aiding with diagonal pressure in the beginning and with vertical pressure at the end. The h5 square is a common place for such queen maneuvers to begin or end. It's an especially useful square because the queen then can also seal off g6 as a flight square, as it does here.

And now you know why it's called Greco’s mate.