I am not only making this covenant and this oath with you … but with those who are not here with us this day. (Deuteronomy 29:13-14)
Moses renews the covenant at Sinai with the Jewish people, which obligated them and all future generations to keep God’s commandments.
Although the Torah permits us to prohibit things or obligate ourselves in matters beyond that which is necessary, there is a custom to perform a general annulment of these personal oaths and vows in front of three lay ‘judges’ before Rosh HaShannah, especially as the specific nature of them may have been forgotten (see Koren Rosh HaShannah Machzor pages 2-5). Maimonides (d. 1207) notes that oaths and vows are taken very seriously in Jewish law and breaching them violates a number of Torah prohibitions (Laws of Vows 1:4).
Although women are certainly permitted to annul their own vows, historically the custom never developed to do so. If a woman has a specific vow which she would like to annul, she should do so in person in the same way that men do before Rosh HaShananh. In this scenario, the Shulchan Aruch (Code of Jewish Law, Yoreh Deah 234:57) ruled that her husband cannot form one of the judges, even though other relatives could.
It is possible however for a man to make the general annulment before Rosh HaShannah on behalf of his wife. Ideally she should explicitly ask her husband to do so, although provided her husband knows that she wants her vows annulled, he may act as her emissary. How is it possible to annul a vow on behalf of someone else?
Rabbeinu Bachaye ben Asher (d. 1340) explains from the verse above that all future generations were also obligated in the covenant at Sinai because future generations retain a spiritual connection to their ancestors just as the branches and leaves of a tree are connected to and sustained by its roots. In that same way that the covenant at Sinai was enacted for all future generations through their intrinsic connection, perhaps hatarat nedarim highlights the spiritual bond between husband and wife which forms them into one unit.