This is one of the essays that I have wrote for my blog that is entitled Steps to Repentance. If you see any errors, please let me know. I am putting this here as I think this would make a good counter-missionary resource to those who say "blood is the only means of atonement."
King Solomon said that there was not a righteous man that does not sin. However, when one sins, we must atone for it and return to G-d. Why would G-d forgive us? Because he loves us and desires to have a relationship with us. G-d says repeatedly:
Yisraél, return to Adonai your God... return to Adonai..." (Hoshé'a 14:2-3)
Seek Adonai when He is found, call Him when He is near.The wicked shall give up his way, and the man of iniquity his thoughts, and he shall return to Adonai, Who shall have mercy upon him, and to Hashem our G-d for He will freely pardon. (Yeshayahu 55).
We know that G-d desires us to repent of our sins (Ezekiel 33), so now, how is it that we do t'shuva?
Step 1: Return from the sin
G-d is perfect; therefore His Law is perfect and unchanging. Because we realized we have sinned and have turned from G-d's Law, we must turn away from the sin. By definition, one cannot possibly do Teshuvah if one commits the same sin purposefully.
Step 2: Regret
Regret is one of the most important steps. In his essay, The Scriptural Position on Atonement, professor Mordochai points out that repentance is a state of mind: i.e, being sorry for the wrongful acts one has committed. It is not the act of being "sorry because you are caught", but genuine remorse--pain that one feels after doing something wrong.
Step 3: Confession before God
The word confess means to agree with. This consists of an oral confession spoken out loud in which one formulates in words the commitments and attitudes one has reached in his or her heart. The oral confession should go along with step 2.
Step 4: Acceptance for the future
It has been said that one knows if they have truly repented if one is placed in the same situation again and does not commit the wrongful act. This consists of resolving (literally a purpose in one's heart) never to commit the sin ever again.
But, what if the act is against a fellow human?
The steps I have listed above only works for sins against God. If one commits sin against another person, one must first ask forgiveness from that person before God will accept the repentance.
Sidebar note: One cannot say "I will sin and repent and sin again", it simply does not work that way. It must be a genuine remorse. This is one thing that sets Judaism apart from Christianity: there is no one quick-fix prayer.
After one has sinned, repentance is one of the most beautiful and most satisfying things one can do. It is important to repent to restore your relationship with fellow man and with Hashem who created us.
(major reference: http://www.ou.org/chagim/elul/foursteps.html)