I basically understand the principle of doubling: using the doubling cube to double the stakes of the current game. I'm not sure when the dice rolled can be double for 4 moves. Is that only when the doubling cube is used? Or can it be done any time within a game? Many thanks!
I think you are confusing two uses of the word double. If you roll doubles (e.g both dice show 3) you move not two but four counters three spaces ('Movement, 3' here).
You can also use the doubling die to increase the stakes at certain times; this has no effect on the game unless your opponent chooses to resign the game rather than play for (and lose) the higher stakes( 'Doubling').