If an effect would create one or more tokens under your control, it creates twice that many of those tokens instead.
If an effect would put one or more counters on a permanent you control, it puts twice that many of those counters on that permanent instead.
Doubling Season affects permanents that enter the battlefield with a certain number of counters.
If there are two Doubling Seasons on the battlefield, then the number of tokens or counters is four times the original number. If there are three on the battlefield, then the number of tokens or counters is eight times the original number, and so on.
Planeswalkers will enter the battlefield with double the normal number of loyalty counters. However, if you activate an ability whose cost has you put loyalty counters on a planeswalker, the number you put on isn't doubled. This is because those counters are put on as a cost, not as an effect.