614.8. Regeneration is a destruction-replacement effect. The word "instead" doesn't appear on the card but is implicit in the definition of regeneration. "Regenerate [permanent]" means "The next time [permanent] would be destroyed this turn, instead remove all damage marked on it and tap it. If it's an attacking or blocking creature, remove it from combat." Abilities that trigger from damage being dealt still trigger even if the permanent regenerates. See rule 701.12.
701.12. Regenerate
701.12a. If the effect of a resolving spell or ability regenerates a permanent, it creates a replacement effect that protects the permanent the next time it would be destroyed this turn. In this case, "Regenerate [permanent]" means "The next time [permanent] would be destroyed this turn, instead remove all damage marked on it and tap it. If it's an attacking or blocking creature, remove it from combat."
701.12b. If the effect of a static ability regenerates a permanent, it replaces destruction with an alternate effect each time that permanent would be destroyed. In this case, "Regenerate [permanent]" means "Instead remove all damage marked on [permanent] and tap it. If it's an attacking or blocking creature, remove it from combat."
701.12c. Neither activating an ability that creates a regeneration shield nor casting a spell that creates a regeneration shield is the same as regenerating a permanent. Effects that say that a permanent can't be regenerated don't prevent such abilities from being activated or such spells from being cast; rather, they prevent regeneration shields from having any effect.
701.6. Destroy
701.6a. To destroy a permanent, move it from the battlefield to its owner's graveyard.
701.6b.
The only ways a permanent can be destroyed are as a result of an effect that uses the word "destroy" or as a result of the state-based actions that check for lethal damage (see rule 704.5g) or damage from a source with deathtouch (see rule 704.5h). If a permanent is put into its owner's graveyard for any other reason, it hasn't been "destroyed."Example:
Doom Blade 
,
Day of Judgement 
,
Shivan Meteor 
, anything with Deathtouch.
701.6c. A regeneration effect replaces a destruction event. See rule 701.12, "Regenerate."
701.14. Sacrifice
701.14a. To sacrifice a permanent, its controller moves it from the battlefield directly to its owner's graveyard. A player can't sacrifice something that isn't a permanent, or something that's a permanent he or she doesn't control.
Sacrificing a permanent doesn't destroy it, so regeneration or other effects that replace destruction
can't affect this action.
704.5f. If a creature has toughness 0 or less, it's put into its owner's graveyard.
Regeneration can't replace this event.
