In short, there is no direct equivalent in ancient Greece or Rome to our specific understanding of the word "sin." But we'll expand a bit. Just for fun :)

Firstly our Christian concept of sin "accomodates two basic and coherent senses: offence against moral codes, and action against the laws or the will of God" (OCD). The closest concept in Greek would be hamartia, meaning something on the lines of "error." Similarly in Latin we have peccatum ("error"), or scelus ("crime"). Peccatum in particular would be latched onto by later Christian authors and developed into something we can more easily equate with "sin", but it's still not the same. Sin requires conscious, voluntary offence, whereas hamartia and peccatum do not. For example: accidentally offending a deity with an improper number of sacrifices would be just as reprehensibly vulnerable to divine retribution; in this way we see why "error" in the very real sense of "human error" plays into ideas such as hamartia and particularly into the Roman peccatum. This is not to be confused with dichotomic entities such as Concordia and Discordia, which were personified values.

That being said, at the heart of Greek and Roman ritual was an ordered ceremony consisting of rites with the purpose of communication or the achieving of goals. There were the most memorable kind of rituals around animal sacrifice, but there was also the offering of cakes and the pouring of libations. To account for errors in a ritualistic setting, there may follow other purification rituals, such as lustration -- deriving from "cleansing" or "washing" or even "shining" in the way that the sun or a fire washes light across a space.

It is only in a ritualistic context, however, that we see similarities in antiquity to the Christian idea of "sin."