The Days of Awe are the ten days between the Day of Judgement (Rosh Hashanah) and the Day of Atonement (Yom Kippur). It is a time when we examine our deeds over the last year and begin a sincere effort to improve our behavior over the next year.
Why does the Day of Atonement come after the Day of Judgment? Wouldn’t it make more sense for God to judge us AFTER we’ve atoned for our sins? One answer comes from Rav Yisroel Salanter (1810-1883), the founder of the Mussar movement of self-improvement. Rav Salanter taught that the purification of Yom Kippur is the result of a long process of changing a person’s character in a very deep way. This process starts with having our priorities straight, and the first priority is acknowledging God as King over all creation. Rosh Hashanah is the critical beginning of the process of truly acknowledging His Kingship over us, an acknowledgement that changes us and leads to the inner cleansing that comes on Yom Kippur.