Some people observe the admirable custom to fast every Erev Rosh Chodesh, which is considered "Yom Kippur Katan," a "miniature" Yom Kippur which is conducive to earning atonement through Teshuva. Similarly, there is a practice to fast every Monday and Thursday during the "Shovemim" period (the weeks during which the first six Parashiyot of the Book of Shemot are read). These fasts are not obligatory, but rather a personal measure of stringency that many people have accepted upon themselves.
According to Halacha, a person who took upon himself such a practice without stipulating "Beli Neder," that he does not accept this practice as a formal vow, and then chooses to discontinue the custom, must first perform "Hatarat Nedarim." This entails standing before three people and explaining to them his vow and the reasons why he feels unable to continue the practice. They then formally declare him absolved of his vow, and he is no longer bound by his acceptance of the given practice.
Some authorities maintain that one does not require Hatarat Nedarim in this case, as he may rely on the declaration customarily made on Erev Rosh Hashanah or Erev Yom Kippur formally renouncing any vows taken during the coming year. According to these authorities, this declaration is effective in canceling any and all vows from the outset. However, as Chacham Ovadia Yosef rules in his work Halichot Olam (vol. 8, p. 59), Halacha follows the stringent view, requiring one to make Hatarat Nedarim if he accepted a religious practice that he now wishes to discontinue.
This Halacha applies only if one wishes to permanently discontinue the practice. However, in a case of a person who normally fasts on Erev Rosh Chodesh but one Erev Rosh Chodesh feels a bit frail or ill and wishes not to fast that day, then, according to Chacham Ovadia Yosef, he does not require Hatarat Nedarim. Since he does not wish to permanently discontinue the practice, he may eat that day without first performing Hatarat Nedarim.
Several pages later (p. 71), Chacham Ovadia discusses the question of whether one may assign somebody else to make Hatarat Nedarim in his stead. Meaning, if somebody wishes to annul his vow or discontinue a religious practice that he has taken upon himself, may he ask his friend to assemble three people and conduct Hatarat Nedarim on his behalf?
Chacham Ovadia distinguishes in this regard between a formal vow that one took upon himself, and the aforementioned case of a person who had grown accustomed to a certain religious practice. In the case of an actual vow, then one cannot appoint somebody else to perform Hatarat Nedarim on his behalf. When, however, a person wishes to discontinue a religious practice, such as fasting on Erev Rosh Chodesh, which he had not formally accepted upon himself as a vow, then his obligation to continue the practice applies only on the level of Rabbinic enactment, as opposed to Torah law. Hence, in this case, one may rely on the authorities who allow one to perform Hatarat Nedarim on somebody else's behalf.
Summary: One who had accepted upon himself a religious practice, such as to fast every Erev Rosh Chodesh, and had not stipulated that he does not accept it as a formal vow, may not permanently discontinue the practice unless he first makes Hatarat Nedarim. In such a case, he may appoint somebody else to perform Hatarat Nedarim on his behalf. Hatarat Nedarim is not required if he wishes to discontinue the practice only temporarily, such as due to illness, frailty and the like. A person who formally utters a vow may not break the vow until he personally performs Hatarat Nedarim; he does not have the option of appointing his fellow to perform the Hatara on his behalf.
