A more recent article on "sin" is here.
This earlier article examines the nouns that are translated as "sin," and "sinner," and the verb translated as "to sin." This article was one part of a larger article about the phrase "forgive sin." That original article is here. The Greek words, like these English ones, are all from the same root, but they do not mean the same things as our word "sin."
No English word can be used in quite the same way. as the Greek word. Though the common word, "mistake," comes closest to the idea in English. The Greek word has the same idea of a "miss," literally meaning "missing the mark" in archery. Translating the root as "mistake" is much wordier in English because we must say "a mistake," "a mistake maker," and "to make a mistake."
The Greek Word and Alternatives
The word translated as “sin” in the Gospels is either the verb, hamartanô, or the noun, hamartia (ἁμαρτία), which mean “to miss the mark,” “to fail in one’s purpose,” “to err,” “to be mistaken,” "to fail in having," “to neglect,” "failure," "fault," and "error." How important was sin in Jesus's teaching? He only used this word in seven verses. Sin may have become a dominant aspect of Christian philosophy, as in the idea that Jesus died for our sins, but it was not a major part of what Jesus talked about.
In the text, Jesus gives us a good idea of his view of this word when he equates it with going into debt. He does this in Luke 13:2 and Luke 13:4. In the first of these verses, Luke 13:2, he uses the adjective form of hamartia, (28 verses) translated as "sinner," hamartolos, (15 verses), which means "the erring" or "the erroneous." However, the word translated as "sinner" later in Luke 13:4 is a different word, opheiletes, which means "debtors." In translating both of these words as "sinner" the KJV translators are recognizing the Jesus is using them both to mean the same type of people in the context of these verses, but what is hidden in that their shared meaning is not a reference to a sin against God, but the idea of making bad decisions, making mistakes leads directly to falling into debt.
This real meaning of hamartia as "missing the mark" is a well-known fact, so much so that many preachers refer to it that way. It is commonly mentioned in modern-day sermons that quote verses referring "sin." However, it is somehow forgotten in translating the NT, where it is almost without exception translated in its different forms as "sin," "sinner," and "sinning."
You should also know that there are well-known Greek words in the era that do mean sin. In Greek, the word that actually means “sin” is alitros (ἀλιτηρός). This word means "sinning against the gods." Different forms mean “sin,” “sinner,” and “sinful.” However, Jesus never uses this term in a form translated as "sin." It is unlikely that Jesus and the Gospel's authors didn’t know this. Words from this base are often used in ancient Greek. We see alitria ("sinfulness"), aliterios ("sinning"), and aleites ("sinner") in ancient Greek. Most amusingly, we even see aletheia, which means "sincerity." The term captures the idea that lying is tied with sinning.
English Translation
The best English translation for hamartia is "a mistake" or "failure." After all, this is the way we say "missing the mark," which is the word's primary meaning. It also works best because the verb most commonly used with the noun hamartia is poieo, which primarily means "to make." The phrase "make mistakes" is a much better translation for what is commonly rendered in Bible translations as "to do sin" or "to practice sin."
This meaning of "mistake" is supported all the other words Jesus says must be "forgiven." For example Jesus uses the Greek word, paratoma, which means "blunders" that we "forgive" in Matthew 6:14 (For if you forgive men their trespasses...) . Again, with this word, there is nor real religious dimension. It is simply a Greek synonym for harartia. Note this is not the same word that some biblical translations translate as the "trespasses" that are "forgiven" in the Lord's Prayer (Matthew 6:12). That word, opheilema, simple means "debts" very much in the financial sense. In a sense, we might infer that debt is a mistake or misstep from Christ's perspective.
A lot of things that Jesus says make much more logical sense when we replace "sin" with "mistake." For example, Christ says in John 8:21 that he is going away and that they would seek him and "die in [their] sins" because they cannot go where he is going. We have to ask ourselves what sin have they committed in seeking him? However, it is easier to understand if Jesus is saying that they will die, "within their failure" because they do not know how to follow. This makes much more obvious sense.