Summary
If someone vows to perform an act of obedience that is not wâjib in Sharia, and this act is within their capacity, then the vow is valid and binding on them.
If someone makes a vow to do something that is impossible, or that is already wâjib in Sharia, the vow is invalid. (If it is wâjib, it still must be done, not because of the vow but because of its original obligation.)
A vow that does not fall under either of these two scenarios is considered valid but not binding (that is, fulfilling it is not required, even though it is legally consequential and requires expiation if not fulfilled), as when one vows to do something that is beyond their capacity or that concerns something not in their possession.
If one vows to do a permissible act, he or she may either do it or expiate. If the act is makrooh, one may still do it or expiate, although it is encouraged to expiate and avoid it. If it is haram, one must avoid it and expiate. If the vow is for lajâj (encouragement or deterrence) or made in a state of anger, one may either fulfill it or expiate. The same is true whenever an oath is intended. Finally, an undefined vow makes an expiation binding. The expiation of any of these vows is like that of an oath.