In standard Minesweeper, the first time you click on a mine, you lose immediately. If you want the game to be easier, you can tell the computer to allow a certain number of mistakes (similar to lives or hit points in other games; this is likely to be considered cheating by some). This is helpful if you accidentally click the wrong square or when there's a place where there's no way to know which square has a mine.
The list of Best Times stores the number of mistakes you've made. If you've made less mistakes than the previous best time, you automatically get the best time, even if you actually took more time than them.
When mistakes are allowed, clicking a square that you've marked as a mine uncovers the square anyways. (Clicking on a square that's been marked as a mine is ignored if mistakes aren't allowed.)
Changes to the number of mistakes allowed don't take effect until the next game.