Hadith: The expiation for a vow is the same as the expiation for an oath

عن عقبة بن عامر رضي الله عنه عن رسول الله صلى الله عليه وسلم قال: «كَفَّارَةُ النَّذْرِ كفارةُ اليَمِين».
[صحيح] - [رواه مسلم]

‘Uqbah ibn ‘Āmir (may Allah be pleased with him) reported that the Messenger of Allah (may Allah's peace and blessings be upon him) said: "The expiation for a vow is the same as the expiation for an oath."
[Authentic hadith] - [Narrated by Muslim]

Explanation

This Hadīth states that the atonement for breaking a vow is the same as the atonement for breaking an oath, which is feeding ten poor people, or providing them with clothes, or emancipating a slave. If one cannot do any of that, he fasts three days. This is the atonement for failing to fulfill a vow. Vows that are meant here are of different types: First, to make an absolute vow. For example, one vows in general terms without naming something, or says that if he does such and such thing, a vow is due on him to Allah. If not fulfilled, this vow must be expiated in the same way an unfulfilled oath is expiated. Second, to make a vow to commit a sin or miss an obligation. This vow must not be fulfilled. In this regard, ‘Ā'ishah (may Allah be pleased with her) reported that the Prophet (may Allah's peace and blessings be upon him) said: ''No vow should be taken to commit a sin, and its expiation is that of an oath.'' Third, to vow to do something that is to too hard to observe, such as a continuous act of physical worship or an exorbitant expense, etc. An expatiation for an oath is to be made in this case. ‘Ā'ishah (may Allah be pleased with her) said concerning a man who vowed to give his money in charity to the needy: ''The expiation of an oath (is due on him).'' Fourth, vowing to do an act of righteousness, such as praying, fasting, performing Hajj or ‘Umrah, with the intention of getting closer to Allah. Such a vow must be fulfilled, whether it is absolute or conditional on the occurrence of a good thing or the removal of a bad thing. For example, one says: if Allah cures my sick relative, or if my wealth that is not with me returns to me safe, then I will do such-and-such (i.e., I will give charity, etc), then he has to fulfill the vow if his condition is met.