to sin: to do what is morally wrong, failing the standard God or another person has a right to expect
This is the New Testament's basic verb for going wrong morally. The picture behind the word is missing a target, like an arrow falling short of the mark, but by the first century that image had faded and the word simply meant doing wrong. To sin is to fail a standard you were meant to meet.
The standard can be named in different ways, and the senses below sort the occurrences by who or what is wronged. Most often the offense is against God or against the moral order he set, whether or not he is mentioned by name. Sometimes the offense is plainly against another person, marked in the Greek by a little phrase meaning "against you" or "against the brothers." A few times the word is closer to "to be at fault" or "to do something blameworthy," as when Paul argues that getting married is no fault at all.
These are shades of one idea, not different words. The same verb covers all of them because in this way of thinking a wrong done to a person is finally also a wrong against God, and a single English word, sin or do wrong, carries the whole range. The writers do not treat sinning as an occasional slip so much as a power people are caught in and need rescuing from.