Quote from Rezzahan »Maybe an illustration of a simple exampe will help your understanding.
Your opponent has the Deathmantle, the Composite Golem, and an Eager Cadet. All his lands are tapped, the Golem is the only source of mana avaialble to him right now. You have Pestilence and enough black mana to activate it multiple times.
1) You activate Pestilence once, and put that on the stack. Then you pass.
2) Your opponent responds by sacrificing the Golem, which gives him 5 mana, and puts a Deathmantle trigger for the Golem on the stack.
701.37a To exert a permanent, you choose to have it not untap during your next untap step.
701.37b A permanent can be exerted even if it’s not tapped or has already been exerted in a turn. If you exert a permanent more than once before your next untap step, each effect causing it not to untap expires during the same untap step.
701.37c An object that isn’t on the battlefield can’t be exerted.
701.37d “You may exert [this creature] as it attacks” is an optional cost to attack (see rule 508.1g). Some objects with this static ability have a triggered ability that triggers “when you do” printed in the same paragraph. These abilities are linked. (See rule 607.2g.)