If a source would deal damage to a permanent or player, it deals double that damage to that permanent or player instead.
Dictate of the Twin Gods applies to any damage, not just combat damage. It also doesn’t matter who controls the source of the damage that’s being dealt.
The source of the damage doesn’t change. A spell that deals damage will specify the source of the damage, often the spell itself. An ability that deals damage will also specify the source of the damage, although the ability itself will never be that source. Often the source of the ability is also the source of the damage.
If more than one Dictate of the Twin Gods is on the battlefield, damage dealt will double for each one (two of them will end up multiplying the damage by four, three of them by eight, and four of them by sixteen).
If multiple effects modify how damage will be dealt, the player being dealt damage or the controller of the permanent being dealt damage chooses the order to apply the effects. For example, the ability of Decorated Griffin says “Prevent the next 1 combat damage that would be dealt to you this turn.” Suppose you would be dealt 3 combat damage and you activate the ability of Decorated Griffin. You can either (a) prevent 1 damage first and then let Dictate of the Twin Gods’s effect double the remaining 2 damage, for a result of being dealt 4 damage, or (b) double the damage to 6 and then prevent 1 damage, for a result of being dealt 5 damage.