If ever a creature's toughness is reduced to zero (or less), it will be destroyed as a state-based effect. If a 1/1 creature gets a -1/-1 counter, then effectively it has a toughness of zero, and will be destroyed.
Edit: Note that a creature having 0 toughness is not a destroy effect. As per the state based actions, it will be put into its owner's graveyard. This does trigger "Dying" effects. This is an important distinction as having 0 toughness is one of the ways to get rid of indestructible creatures.