You know it’s good practice to castle: it brings your rook out from the corner and it puts your king safely behind a wall of pawns. Unfortunately your opponent knows this too; and if he does castle, how are you supposed to get at his king? Every game is different, but there are two general answers to that question. One, which we just examined, is to attack along the back rank, using the pawns in front of the enemy king to your advantage as they prevent it from escaping your rook or queen. The other general answer is to attack from the top and from an angle at the same time.
There is a large family of mating patterns that involve this latter principle: establishing two lines of pressure, one diagonal and one vertical, from which the castled king cannot escape. One of the lines of pressure typically is exerted by a bishop aimed into the king’s corner; the second, vertical line of pressure is exerted by a rook or queen along the h-file or g-file. The rook or queen may end up on a square next to the enemy king or may do their work from a distance. As we shall see, sometimes a knight or a second bishop or even a pawn can do the work that normally would be done by one of those other pieces just mentioned; but the easiest way to get started is just to think in terms of combining diagonal (bishop) and vertical (queen/rook) pressure in various ways.
The usual challenge in these mates is to establish those two lines of pressure by tearing open the enemy king’s pawn cover with a sacrifice. The castled king usually starts with a nice row of three pawns in front of it. If one of those pawns steps forward even one square, it creates a vulnerability: an open diagonal leading into the king’s position. This is especially true of the g-pawn and f-pawn, as in their starting positions they block the diagonals leading to h8 and g8 respectively. So when a pawn in front of the castled king moves, consider it an invitation to place a bishop so that it runs through the square the pawn used to occupy.
Then to create the complementary vertical pressure you need to open a file leading to the enemy king. This means the enemy pawn occupying the file must be removed so that your rook or queen will have a straight shot toward the same square or sector that your bishop attacks. There are various ways to do this. The important point for now is the overall goal: disrupting the pawn cover in front of the king to allow coordinated attacks against its position.
Many of these mating patterns are beautiful and have storied pedigrees, being named after inventors or popularizers who lived long ago. The explanation of each will start with a skeletal example showing how it looks with the non-essential pieces cleared off the board. The key ideas and squares are identified, usually from White’s side; thus we might speak of the importance of the f7 square, with the understanding that if you are playing the Black pieces you use the square comparable to f7 from your side (i.e., f2). Sometimes we will avoid this confusion by referring to the “king’s bishop”—the bishop that starts the game to the right of your king. If you are playing White, this is your light-squared bishop; if you are playing Black, it is your dark-squared bishop.
One more bit of jargon: we often will speak of “heavy pieces,” which (in case you have forgotten) means either a rook or queen. It is convenient to have a way to refer to those pieces without distinguishing between them; there are many mating functions that can be filled equally by either a queen or rook, and which you use will depend on which is available to you.