Why is ELUL a time of Forgiveness?
Moshe went up to Har Sinai to receive the Torah (included in that was the tablets with the 10m commandments). The Jewish people misunderstood when he was supposed to come back. Having relied on him as a intermediary of sorts, they panicked without him and rushed to make a substitute for him (a golden calf-(don't ask here what it really was) to relate to God. Moshe came down and saw and broke the tablets. Hashem was sort of done with the Jewish people and offered to start all over again with Moshe. Our hero could have said to himself (wow the entire Jewish people rebuilt form me) - but he didn't. Our teacher, Rabbi and hero pleaded to Hashem not to destroy us. BTW Moshe went up to receive the Torah on the 6th of Sivan and this sin occurred on the 17th day of Tammuz. (a fast day now to repent even today)
Moshe prayed and begged Hashem not to destroy the Jewish people for the next 40 days. Please take them back. On Rosh Chodesh ELUL -which today is the anniversary, Hashem said Okay -I forgive you enough not to destroy you but you have to repair the damage to the relationship that was done. When we hurt one another we not only did or say something to hurt the other person but in addition we have created a separation, a chasm, distance between one another-even if we forgive someone or they us -that is the easy part - ill hurt feelings still remain. We now have to work to be close again. Here too Hashem forgave us on Rosh Chodesh ELUL but the relationship was damaged.
These next 40 days are rebuilding our relationship with Hashem through repentant, tefillah and Tzeduckah.