Gemara

21b (Rav Yosef citing Rav Asi): A Chacham can permit a vow only if it is like one of the four vows.

He holds that we are not Pose'ach a vow with regret.

A man who had vowed came to Rav Huna to permit the vow.

Rav Huna: Do you still desire the vow?

The man said 'no', and Rav Huna permitted the vow.

A man who had vowed came to Rabah bar Rav Huna to permit the vow.

Rabah bar Rav Huna: If 10 people had appeased you at the time of the vow, would you have vowed?

The man said 'no', and Rabah bar Rav Huna permitted the vow.

(Beraisa - R. Yehudah): If one seeks permission of a vow, we ask if he still desires the vow. If he says 'no', we permit it;

R. Yishmael b'Rebbi Yosi says, in the name of his father, we ask him 'if 10 people had appeased you at the time of the vow, would you have vowed? If he says no, we permit the vow.

A man came to R. Asi to permit a vow. R. Asi asked him if he regrets it,

The man (incredulously): No?! (Of course I do!)

R. Asi permitted the vow.

22b (Rava): The Halachah is, we are Pose'ach with regret.

Rishonim

The Rif brings our Gemara with minor changes.

Rosh (3:1,2): Rav Ashi (our text - R. Asi) permitted only if the Noder regretted that he ever vowed. Some texts say 'if you were happy with the vow until now, but now you regret, we do not permit.' The Noder must be sure that he regrets that he ever vowed. If not, it is not permitted, and he is forbidden forever.

Ran (21b DH Omar Lei Lo): If regret from now were enough, why did Rav Huna ask the Noder if he wanted the vow? Surely, he came to permit it because now he regrets it!

Rambam (Hilchos Shevu'os 6:5): To permit an oath, one comes to a Chacham and says 'I made this oath, and I regret it. Had I know that I would be in such pain, or that this would occur, I would not have sworn. Had my mind been then like it is now, I would not have sworn.' The Chacham asks if he regrets; the Nishba says 'yes', and the Chacham permits him.

Rambam (Hilchos Nedarim 4:5): If one vowed and regretted it, he asks a Chacham, who permits it. Permission of vows is like permission of Shevu'os

Ran (DH Ilu): Rabah bar Rav Huna asked 'if 10 people had appeased you at the time of the vow, would you have vowed?' Some texts say that he holds that we are not Pose'ach with regret. Other texts say that he holds that we are Pose'ach with regret. Both mean the same. 'If 10 people had appeased you' is worse than a Pesach, but it is better than regret. It is worse than a Pesach, for we do not have 10 people now, which would allow us to say 'had you known that these people would appease you...' It is better than regret, in which he does not find anything that would have deterred him at the time. The Pesach of 10 people shows that he vowed out of haste.

Beis Yosef (YD 228 DH u'Mah she'Chosav u'Keitzad): Perush ha'Rosh (21b DH Ilu) connotes that the Pesach of 10 people works only if there are 10 people appeasing him now.

Ran (22b DH Nodarta): (Rav Schorah came before Rav Nachman with a vow to permit. Rav Nachman strove to find a Pesach, but failed; he did not permit the vow.) Even though Rav Nachman holds that we permit with regret, he knew that Rav Schorah was stringent not to permit through regret. Some say that he had vowed to fast several fasts, and he did not regret from the beginning, lest he forfeit the reward for what he already fasted.

Poskim

Shulchan Aruch (YD 228:7): One who regrets the Neder itself does not need a Pesach. It suffices that he wishes that he never vowed. If he was happy with the vow until now, but now he regrets, we do not permit. The Noder must be sure that he regrets that he ever vowed. If not, it is not permitted, and he is forbidden forever.

Rashba (Teshuvah (attributed the Ramban) 246, cited in Beis Yosef DH ubi'Teshuvas): If a sick person vowed never to eat cheese because it aggravates his illness, and now he regrets it, we cannot permit through this. At the time he vowed he needed the vow. Only now he desires cheese and regrets it! A different Pesach is needed.

Shulchan Aruch (ibid.): The Pesach can be from the Neder itself. E.g., he said 'Ploni may not benefit from my property if he does this act', and Ploni did it. We ask, 'had you known that he would do it, would you have vowed?' If he says 'no', we permit him.

Rashba (Teshuvah (Ramban) 255, cited in Beis Yosef DH uv'Siman): A man vowed not to eat meat and wine if he will transgress a certain Aveirah. He says that he regrets from the beginning, but this is only because he transgressed. If not, he would want the vow. We Pose'ach for him 'had you realized that the Yetzer Ra would overpower you, would you have vowed?' This is like Rav Yosef, who was Pose'ach for Abaye 'had you known that your wife would transgress your will (and cause your vow to take effect), would you have vowed?' One could distinguish based on whether the vow depends on someone else or on himself, but it seems that they are the same, for the Yetzer Ra is prone to overcome people, especially if he often transgresses this. It seems that in this case, even regret from now and onwards suffices. Regret from the beginning is required only for vows concerning the matter itself, e.g. one who wants to fence himself from wine or something that harms him. These are strong vows, so they require regret from the beginning. A vow that is a fine is lighter. He did not want to forbid the subject of the vow. Regret from now and onwards suffices.

Mordechai (cited in Beis Yosef DH veha'Mordechai): R. Yosef Bechor Shor says that there are three regrets. All agree that we permit with something that already occurred, and had the Noder known he would not have vowed. Nolad, i.e. a totally unexpected development after the Neder, cannot be used to permit. Even if the person considered the possibility that the Nolad will occur, he would not have refrained due to this. Amora'im argue about regret due to a common development after the Neder. The stringent opinion considers this like Nolad. It is good to be stringent, like R. Yosef Bechor Shor.