In "The Cask of Amontillado", what does Montresor mean when he says he seeks to "punish with impunity"?
Montresor begins his narration of "The Cask of Amontillado" by introducing us to his hatred of Fortunato, though he fails to explain exactly what Fortunato did to engender this hatred. Montresor establishes that these injustices must be addressed, but as he is a calculating and intelligent person, he intends to do so in a careful and dispassionate manner, lest his anger get the better of him and tarnish his justice with something so inconvenient as an arrest and incarceration.
By "impunity" Montresor means to put himself beyond all suspicion or reproach, yet he still seeks to attack Fortunato in a very specific way. It must be clear to Fortunato that he is being punished, that the punishment comes from Montresor, and that the punishment is in return for Fortunato's insults. However, one might imagine that it would be difficult to accomplish this without committing some crime to which Montresor could be traced; thus his need for impunity, i.e. some way to ensure that there are no witnesses, no evidence, and no means of connecting Montresor to the crime.
This is fulfilled by the crime being committed in an empty home, and the only witness (Fortunato) dying along with it. Montresor never mentions being held suspect for Fortunato's disappearance, just as he planned.
Montresor means that he plans to suffer no consequences for his crime when he notes he will "punish with impunity." He has carefully thought out every detail of his plot to get revenge on Fortunato, from whom he believes he has suffered a "thousand" injuries. Therefore, he waits for the noisy and drunken revelries of the Mardi Gras before he acts, knowing neither he or Fortunato will be missed, and he plays on the vanities of Fortunato to lure him to a damp, dark catacomb beneath his home. Montresor's servants are gone and nobody is around to witness him walling up his enemy nor to hear Fortunato's screams. It has worked out to be the perfect crime, as fifty years later Montresor, still unsuspected, is telling the story, seemingly as a deathbed confession.