In checkers, what is supposed to happen when a piece gets to the other side of the board and becomes a king? Does the move end? Can the piece continue to skip in the same turn it becomes a king? Can it choose not to be kinged and skip again? I've looked up a couple of different rules and I can't really find any sort of set rules for this. What is the actual rule?
From the Wikipedia article on checkers:
If a player's piece jumps into the kings row, the current move terminates; having just been crowned, the piece cannot continue on by jumping back out (as in a multiple jump), until the next move.
There are many variants of checkers/draughts, and for most of them (including the Checkers variant played predominantly in the English-speaking world), promotion is both mandatory and ending the move. Quote from the American Checkers Federation's official rulebook:
1.16 When a man reaches the farthest row forward (known as the “king-row” or “crown-head”) it becomes a king, and this completes the turn of play. The man can be crowned by either player ( ) by placing a man of the same colour on top of it before the next move is made. (It may be necessary to borrow from another set if no captured man is available for the purpose).
1.19 If a jump creates an immediate further capturing opportunity, then the capturing move of the piece (man or king) is continued until all the jumps are completed. The only exception is that if a man reaches the king-row by means of a capturing move it then becomes a king but may not make any further jumps until their opponent has moved. At the end of the capturing sequence, all captured pieces are removed from the board.
However, for the variant for Russian checkers, promotion during a capture sequence happens en-passant (i.e. in passing) and the crowned man has to continue jumping as a king.
When a checker reaches the furthest row on the opposite side of the board, its turn ends, because it can make no further FORWARD move. It cannot "skip" or make any other move.
Beginning with the NEXT move, it can move backward. In order to do so, it must be "kinged," which it cannot refuse such, at the end of the preceding turn.