I will not try to provide a definitive answer on this. I will, however, try to provide some basic material; cmkeller, David B, etc., can provide corrections and explanations as they think necessary.
“Sin” (in Hebrew het, “missing the mark”), simply, is violation of the Law (Written and Oral Torah). Halakhah (the Way or the Path), although it has many of the aspects of law, is actually the practical guide to what the Law means (what is “honoring Shabbat and keeping it holy”). Halakhic sages have found 613 mitzvot, or commandmants, in the Law (the notorious “Ten Commandmants” are among them, but have no greater or lesser stature than the other 603). Since even the Holy One, blessed be He, has no authority to introduce new commandmants (there’s a subtle bit of reasoning in arriving at this conclusion), we can say with assurance that if a Jew does follow the mitzvot, he or she is free of sin.
(Different rules are in effect for non-Jews. It is considered by Jewish authorities that a non-Jew need only follow the Noachide laws (exactly how many there are depends on what is considered a “law”, and what is a consequence of a law, but seven is the generally accepted number) to have a part in the World to Come.)
If a sin is committed, it is necessary to ask forgiveness from the Holy One, blessed be He, in an act of atonement. A proper act has the three parts of confession (“I did ‘X’”), repentance (“I’m sorry that I did ‘X’”), and resolution (“I will not do ‘X’ again”). Public confession is not necessary, although it may be thought advisable if the sinful act is publicly known, or directed harmed another person. Likewise, no intermediary with Heaven is required or, indeed, is acceptable, although again the penitent may find the aid of such persons helpful. Restitution may or may not be considered evidence of repentance, depending on the sin.
Previously, the act of sacrifice removed the threat of punishment for sin. Except in a very few cases, sacrifice did not atone for the sin per se. Since the sacrificial service is in abeyance until the building of the Third Temple, it is not now required.
“Kings die, and leave their crowns to their sons. Shmuel HaKatan took all the treasures in the world, and went away.”