lets start with the concept of sin:
The Jewish concept of sin is pretty different to that of Christianity. The word for sin is “chait”. Chait has an alternative meaning- an archery target. So, how do the two meanings interrelate? Literally, sinning is seen like an arrow missing a target. We are given a blueprint in the Torah which paints the target for us- our behaviour determines where on the target the arrow will hit- the more we behave incorrectly, the further from the center it lands.
This ties in to the concept that everyone has a share in the world to come - except for the few that get punished with the punishment of Kares (spiritual excission). The Jews have the Torah to follow- the non-Jews the Seven Noachide commandments. The better we follow the blueprint, the closer to the center we get. Its practically impossible to get the very center down to the last molecule- but then its not necessary, we only have to land on the target to enter the world to come (in this analogy Kares is off the target). Some extend the analogy- they compare the seven levels of the world to come outside of the courtyard of God’s abode described in Masechta Chagigah to the rings on the target. (What is in the courtyard is unknown- at that point the Talmud says “Beyond this point we may not enquire” The Zohar or Sefer Yetsira may have more on this, but I haven’t studied either)) The laws we follow become stabilising influences on the arrow guiding its path. But here is where the Jew is more fortunate than the non-Jew- because the non-Jew only has seven potential stabilisers, hitting the center is much harder, requiring far more perfection on each one for it to do its job properly. Jews, have 613 potential stabilisers- meaning that even imperfect observance of the many facets will be effective in guiding the arrow closer to the center.
In the end, sin is seen as something that moves us off target, damages our status in the world to come by drawing us away from the centre, the direct presence of God.
Now, as for forgiveness:
As for asking for forgiveness- simple:
repentance- truly repenting whatever you did and resolving to never do it again
prayer for asking forgiveness from God
retitution to the person you wronged or charity if you acted against the community.
even in the days of the temple the sacrifice was for sins done accidently (korban chatas) or if you were in doubt of you had sinned in a serious manner (kornab asham tali). For a deliberate sin there was no sacrifice though many woudl bring a korban orlah as a voluntary offering when they repented such a deed. The most important aspect of any sacrifice was the INTENTION behind it- no proper repentance, then no amount of sacrifces would help! Additionally- you could only bring a sacrifice for yourself- nobody could make a sacrifice for you- remember what Is aid about repentance? Ony you can repent for yourself- nobody can do it for you! (the sacrifices made by the Kohen Gadol on Yom Kippur, such as the scape goat sacrifice were for COMMUNAL sins, not individual sins. The thread changing colour was for the community- not the individual- see the Talmud, masechta Yoma)
My blog; orthodox jew