There are four basic checkmates in Chess: King and Rook, King and Queen, King and Two Bishops, King and Bishop/Knight. I believe it to be important to know and understand these basic checkmates. Too many times, I have seen games where one side had these pieces and the othe side having only a King. With proper knowledge, checkmate is inevitable. I have seen players with rankings higher than mine not posess the knowledge of these four basic checkmates, and chase the enemy King needlessly all over the board, or worse yet, blunder into a draw!
"Hey, Captain", you may say. "Why aren't we starting with the King and Queen? The Queen is the most powerful piece on the board! Wouldn't that be a simpler place to start?"
Well, yes and no. True, the Queen is more powerful than the Rook, and there in lies the problem. The Rook covers only ranks and files where the Queen covers Ranks and Files, but also diagonals. With King and Queen, the chances of stalemate is present. Thus, the Queen/King checkmate is a little bit trickier. So, let's start with King/Rook where we don't have to worry about diagonals or stalemates. The same techniques used to checkmate with King/Rook can also be used with King/Queen; we only need to be sure that, until that final checkmate move, our advesary's King has a place to move.
The end result of the King/Rook checkmate looks like this:
The Rook can not be captured, and due to the rule that the Kings must remain one square away from each other, Black can not move out of check.
Getting our advesary's King in this position is not difficult.
The diagram below illustrates the basic King/Rook checkmate:
Thus, the Rook is kept out of reach from any threat from the opponent's King. Our King prevents the opponent King from moving in any direction except backwards. With this principle, it is important to remember that putting one King in front of the other King is the job of your opponent! The Rook and King work together to systematically press the opponent's King towards the edge of the board where the opponent's King can no longer retreat.
With this knowledge, if we ever find ourselves in a position similar to this, checkmate of our opponent is inevitable.