As a spell or ability an opponent controls resolves, if it would force you to sacrifice a permanent (as the annihilator ability does), you just don’t. That part of the effect does nothing. If that spell or ability gives you the option to sacrifice a permanent (as Prowling Pangolin’s ability does), you can’t take that option.

If a spell or ability an opponent controls instructs you to perform an action unless you sacrifice a permanent (as Ogre Marauder does), you can’t choose to sacrifice a permanent. You must perform the action. On the other hand, if a spell or ability an opponent controls instructs you to sacrifice a permanent unless you perform an action (as Rishadan Brigand does), you can choose whether or not to perform the action. If you don’t perform the action, nothing happens, since you can’t sacrifice any permanents.

You may still sacrifice permanents to pay the costs of spells you cast and abilities you activate, or because a resolving spell or ability you control instructs or allows you to do so. Notably, you may still sacrifice Eldrazi Spawn creature tokens for mana.

You may sacrifice a permanent to pay the activation cost of an ability, even if that ability comes from a permanent an opponent controls (such as Excavation).

You may sacrifice a permanent as a special action, even if the effect that allows you to do so comes from an opponent’s permanent (such as Damping Engine or Volrath’s Curse). No one controls special actions.

This ability affects only sacrifices. It won’t stop a creature from being put into the graveyard due to lethal damage or having 0 toughness, and it won’t stop a permanent from being put into the graveyard due to the “legend rule” or the “planeswalker uniqueness rule.” None of these are sacrifices; they’re the result of game rules.