Between Rosh HaShanah and Yom Kippur are ten days, known as the Aseret Yemei Teshuva. During this time it is "exceedingly appropriate" for Jews to practice "Teshuvah", which is examining one's deeds and repenting for sins committed against both G-d and one's fellow man in anticipation of Yom Kippur. This repentance can take the form of additional supplications, confessing one's deeds before G-d, fasting, and self-reflection. On the third day, the Fast of Gedalia is celebrated.
(Adapted from Wikipedia)