The mating of a king with minimal material.
Sometimes a player has a chance to reach a position where there is just a rook. Instead of trying to reach that position they instead go the murky rout in an attempt to reach an even greater advantage. Why? they might not confident that they can mate the king in 50 moves. Perhaps they want to end with more flare. This section is the answer to the first question!
Take this starting position:


White wants to get his king directly across from black, but have it whites turn. White can not simply move his king to c4, though he is across from blacks king, it is blacks turn. So white throws a move away.

Now white shows his superiority, Black has no choice but to cede the e-file. One level closer to the edge of the board.

Black in his effort to stay away from coming across whites king has made another mistake, he is now one level closer to the side bottom of the board. White realizing this, changes so that black will be mated along the bottom.

The end is near, black has been forced back to a side. Black needed to hold of mate for 50 moves to reach a draw by 50 move rule, but it is less then 20 and black does not have a chance.

You see there is no need to fear a rook vs. king ending. If you see a way for force this, then you see a way to force mate!!