Answer:
   
   The way in which atonement is achieved is one of the defining
   characteristics of Judaism. When the temple was still in existance,
   sins were atoned through offerings and the Jewish court system. Since
   the Temple was destroyed, a rabbinic system that has focused on prayer
   has arisen.
   
   The basic philosophy is that for sins against another human, one must
   atone to that person; for sins against G-d, one must atone to G-d.
   Typically, the atonement to G-d occurs on the holy day of Yom Kippur,
   when one prays and repents, and presumably changes one's ways. During
   this time, one also apologizes to those harmed for any grievences,
   intended or unintended. Apologies, however, are not enought. There are
   actually three phases:
    1. Abandoning the sin
    2. Regret
    3. Verbal confession. Thoughts aren't as powerful as hearing yourself
       vocalize them
       
   Note that, in Jewish thought, any sin can be atoned through severe
   repentence, without death.
   
   However, there is the notion that true remorse for certain sins (such
   as murder) can only come with experiencing Yom Kippur, or sometimes
   even only with death. This is particularly true for those sins that
   would have been punished by death in the days of a Jewish court system
   (assuming all the legal details were met). However, that rule isn't
   hard and fast. Just as we can acheive atonement today without Temple
   sacrifices, remorse is possible even without death. There is the
   thought that there are some sins (murder, idolotry, and adultery) that
   one ought not violate even at risk to your own life. For example,
   under threat of death, one might eat non-Kosher food; however, if the
   choice is praying to an idol or death, one is supposed to choose
   death.