Fast Days for Repentance & Atonement
The Jewish calendar has a number of such days in addition to Yom Kippur.
Excerpted from A Guide to Jewish Prayer with permission of the publisher, Schocken Books.
Various days of the year have been instituted as fast days for the purposes of repentance and atonement for sins.
Among these days are to be included the Monday, Thursday, and Monday following the Sukkot and Pesach festivals. These days were decreed as fast days because, as stated in the Talmud (Kiddushin 81a), the festivals, which are days of joy and leisure, may also result in transgressing the limits of responsible conduct. Consequently, these fasts were supposed to have been held immediately after the festivals, in order to atone for any such laxity. But since the months of Tishrei and Nisan are considered joyful months, and it is customary not to fast during them, the Monday-Thursday-Monday fasts are held during the following months--Heshvan and Iyar.
There are further reasons for these fasts: During Heshvan one fasts to plead for rain to fall on the newly planted fields, and during Iyar so that the harvest should not be ruined by blight or mildew. Heshvan and Iyar are also periods of seasonal changes in weather; hence, one fasts and prays for physical health.
Since these days were fixed as public fasts (although not everyone observes them), there are special selihot [penitential prayers] for them also. There is a Torah reading in Minhah, and daily prayers are the same as on any other public fast days.
Fasts Related to the Book of Exodus
Some people fast on the Thursdays of those weeks during which the first eight portions of the book of Exodus (whose initial letters spell out, in Hebrew, the appellation Shovavim Tat) are read. This custom originated among Rabbi Isaac Luria and his disciples, the point of it being that this period is a propitious time for purifying the soul from sin. The period was also set aside for fast days, since it generally occurs during the months of Tevet, Shvat, and the beginning of Adar, during which there are virtually no days on which it is forbidden to fast (except for the 15th of Shvat).
Moreover, these are winter months, during which it is less difficult to fast because the days are cold and short. There are also brief selihot for these days, with different selections for each week.
Yom Kippur Katan: A Minor Day of Atonement
The noted kabbalist [mystic] Rabbi Moses Cordovero of Safed (1522-1579) ordained that the eve of Rosh Hodesh [the new month] be observed as a minor day of atonement--a fast day of repentance and self-purification. Although this custom was not made mandatory, it has been widely accepted among [some] Jewish congregations and is treated as a public fast day. Special selihot are recited after Minhah [the afternoon service]. If the eve of Rosh Hodeshfalls on a Shabbat or Friday, then Yom Kippur Katan is brought forward to Thursday.