Rosh Chodesh is a time of atonement, as we say in the Musaf prayer, “A time of atonement for all [the Jewish people’s] offspring.” In addition, when the Temple stood, a he-goat would be sacrificed as a sin offering.
In order to make this atonement complete, pious Jews customarily repent in preparation for Rosh Chodesh. Some people fast on the eve of Rosh Chodesh and recite special “Yom Kippur Katan” prayers prior to the Mincha service. The day before Rosh Chodesh is called “Yom Kippur Katan” ( Minor Day of Atonement) because it is a time to atone for the sins of the previous month, just as Yom Kippur is a time to atone for the sins of the previous year (see Mishna Berura 417:4, Kaf HaChaim 10-21). Nowadays, few people follow the custom of fasting; instead, one should study more Torah and give extra charity.