What's the Difference Between Triggered and Activated Abilities?

Unlike activated abilities, triggered abilities do not need to be activated, — they trigger regardless of players' wishes and are subject to no limitations. 603.2a. Because they aren't cast or activated, triggered abilities can trigger even when it isn't legal to cast spells and activate abilities.

What is an activated and triggered abilities?

Triggered abilities are put onto the stack the next time a player would receive priority after the trigger event occurred. Activated abilities must manually be activated. Mana abilities may be either activated or triggered, and they don't use the stack.

What happens when you counter a triggered ability?

701.5a To counter a spell or ability means to cancel it, removing it from the stack. It doesn't resolve and none of its effects occur. A countered spell is put into its owner's graveyard.