The Ten Days of Repentance (Hebrew: עשרת ימי תשובה, Aseret Yemei Teshuva) are the first ten days of the Hebrew month of Tishrei, usually sometime in the month of September, beginning with the Jewish holiday of Rosh Hashanah and ending with the conclusion of Yom Kippur.
During this time it is considered appropriate for Jews to practice Teshuvah (literally: "returning" or "repentance") which is examining one's ways, engaging in repentance and the improvement of their ways in anticipation of Yom Kippur. A "penitent" is referred to as a baal teshuva ("master [of] repentance"). This repentance can be expressed in early morning prayers, known as selichot, which capture the penitential spirit appropriate to the occasion and charity, acts of Hesed ("loving-kindness"), or self-reflection.
The first two days of a Ten Days of Repentance are on Rosh Hashanah. One of those days may occur on a Shabbat as well, making that day of Rosh Hashanah on which Shabbat occurs stricter in observance, meaning the observances of Shabbat are followed than a Rosh Hashanah that occurs on any other day but Shabbat (Saturday). When Rosh Hashanah occurs on a Shabbat a few additional prayers in the mahzor ("prayer book") are added as well as excluded in keeping with the combined theme of a Rosh Hashanah and Shabbat combination.
The third day is always Fast of Gedalia, it follows Rosh HaShanah. It is a half day fast, meaning it is only observed from dawn of the third day until dusk of that same day.
After Rosh Hashanah ends and before Yom Kippur starts the next notable day is the special Shabbat that has its own name Shabbat Shuvah ("Sabbath [of] Return") meaning the Sabbath devoted to "teshuva" which means "repentance" in Judaism.