Gemara

(Beraisa): It says "Zeh ha'Davar" regarding vows, and also regarding Shechutei Chutz (slaughtering a Korban outside the Mikdash). Just like Shechutei Chutz applies equally to Aharon, his sons and all of Yisrael, the same applies to vows.

(Rav Acha bar Yakov): This teaches that a Beis Din of three commoners can permit vows.

Objection: It says "Roshei ha'Matos (the heads of the Shevatim)"!

Answer (Rav Chisda): That teaches that a Mumcheh (expert Chacham) can permit by himself.

(Beraisa - Ben Azai): "Moshe told Mo'adei Hash-m to Benei Yisrael" - the Yomim Tovim are mentioned, but Parashas Nedarim was not said with them.

(Rav Sheshes): Ben Azai teaches that the Yomim Tovim depend on (Kidush Rosh Chodesh by) expert judges, but a Beis Din of commoners can permit vows;

"Roshei ha'Matos" teaches that a Mumcheh can permit vows by himself.

Bechoros 36b (R. Chiya bar Amram): Three commoners can permit a Mum in a place where there is no Mumcheh. Three commoners can permit a vow in a place where there is no Mumcheh.

The Seifa teaches that the Halachah does not follow R. Yehudah.

(Beraisa): Three people may permit a vow;

R. Yehudah says, one of them must be a Chacham.

Question: R. Chiya allows three commoners to permit a Bechor where there is no Chacham. What kind of Chacham does he refer to?

Answer (Rav Nachman): He refers to a Chacham like myself.

(Beraisa - R. Yehudah): One of the three (to permit a vow) must be a Chacham.

Question: May the others be utterly unlearned?!

Answer (Ravina): No, they must be able to understand when something is explained to them.

Rishonim

The Rif (24b) brings the Gemara in Nedarim.

Rambam (Hilchos Shevu'os 6:1): If one swore Shevu'as Bituy and regrets it, he asks a Chacham to permit it, or three commoners where there is no Chacham.

Hagahos Maimoniyos (1): The commoners must be able to understand if something is explained to them, like it says in Bechoros.

Rebuttal (Kesef Mishneh): The Rambam does not require commoners who can reason. He explains that only R. Yehudah requires this, for it is unreasonable that Amei ha'Aretz could join with a Chacham. Nedarim 78a supports him. Just like Shechutei Chutz applies equally... to all of Yisrael, the same applies to vows. Commoners for Shechutei Chutz need not be learned! If the Rambam required that they can understand, he would have said so!

Ran (78a DH Omar): Surely, the Hekesh to Shechutei Chutz does not totally equate them. If the three do not understand at all, their Heter is meaningless!

Ramah (Bava Basra 120b 64): The commoners must know Hilchos Nedarim, like in the days of Moshe, when all Yisrael learned from Moshe.

Rambam (Hilchos Nedarim 4:5): Hataras Nedarim is like Hataras Shevu'os. Only a great Chacham can permit, or three commoners where there is no Chacham.

Rashba (78b DH Omar): We learn from Ben Azai that a Mumcheh can permit by himself only if he is Kosher for Kidush ha'Chodesh, i.e. he has Semichah.

Nimukei Yosef (DH b'Yachid): The Rambam says that a Mumcheh can permit. Semichah is not required. The Ro'oh says that the Ramban agrees.

Note: The Ran (23a DH v'Heter) says that the Ramban requires Semichah.

Ran (8b DH v'Shamta): We do not require Semichah. It does not say "Elohim" regarding Hataras Nedarim. A great Chacham fluent in Hilchos Nedarim suffices. The Yerushalmi says that Rav Huna permitted, even though he did not have Semichah.

Rosh (3:3): The three commoners need not be learned or know how to reason, but if something is explained to them, they understand. If three commoners permitted in a place where there is a Chacham, it is permitted. One must go to a Chacham merely for the Chacham's honor. Nowadays, no Chacham can permit a vow by himself. The Yerushalmi says that three who know how to be Pose'ach (find a reason to consider the vow a mistake) can permit like a Chacham.

Beis Yosef (YD 228 DH Keitzad): The Rosh brought the Yerushalmi to teach that we require knowing how to be Pose'ach in addition to ability to reason.

Nimukei Yosef (DH Gemara): The commoners must know how to be Pose'ach or be able to reason. If not, they are not Bnei Torah, rather, like people who sit in the corner.

Poskim

Shulchan Aruch (YD 228:1): If one vowed and regrets it, he goes to a Chacham Mumcheh who has learned and can reason. If there is no such Chacham, he goes to three commoners who understand when one explains to them, and know how to be Pose'ach.

Beis Yosef (DH v'Ta'amo and DH u'Mah she'Chosav v'Ha'idna: The Rosh holds that R. Yehudah and the first Tana argue only about whether or not one of the three must be a Chacham. They do not argue about which commoners may be used.

Drishah (1): This is like monetary cases, in which the judges must understand if one explains to them (CM 3:1). Also, one of them must be learned. The Tur did not specify this. He holds like the Rosh; we assume that one of them is learned. Indeed, if none of the three is learned, they cannot permit.

Rebuttal (Shach 2): Only monetary cases require a learned judge! Hataras Nedarim is not a monetary case! The Levush agrees. This is the practice.

Taz (2): Even though we permit through regret, we require commoners who know how to be Pose'ach, for if they will be able to do so, this is better.

Shulchan Aruch (ibid.): Nowadays, no expert can permit by himself.

Beis Yosef (DH u'Mah she'Chosav v'Ha'idna): The Rosh says that nowadays no expert can permit by himself because no one is as learned as Rav Nachman. The Ran says that the Ramban requires someone with Semichah, which no one has today. The Rambam requires a great Chacham. He holds that Rav Nachman only requires a great Chacham of the generation.

Taz (3): If someone permitted by himself nowadays, it is not permitted.