If you target a creature with 1 toughness, Puncture Bolt deals 1 damage to it, then puts a -1/-1 counter on it, which reduces it to 0 toughness. The creature will be put into its owner’s graveyard as a result of having 0 toughness. Regeneration can’t save it.

Puncture Bolt will resolve fully (and give the creature a -1/-1 counter) even if the 1 damage would be enough to destroy it. (This would matter if a 3/3 creature with persist had already been dealt 2 damage, for example.)

If you target a creature with 2 toughness, Puncture Bolt deals 1 damage to it, then puts a -1/-1 counter on it, which reduces it to 1 toughness. The creature will be destroyed as a result of lethal damage. Regeneration could save it.