Gemara

R. Chiya bar Ami's slave immersed a Nochris (to be Metaher her from Tum'as Nidah) because he wanted to take her for a wife.

Rav Yosef: I can show that she and her daughter are Kesheros:

She is Kesherah, for it is clear from Rav Asi (brought immediately below) that Tevilas Nidah suffices for conversion.

Her daughter is Kesherah, for if a Nochri or slave fathered a child from a Bas Yisrael, the child is Kosher.

People were calling a certain man the son of a Nochris.

Rav Asi: Didn't his mother immerse after being a Nidah?!

People were calling a certain man the son of a Nochri.

R. Yehoshua ben Levi: Didn't his father immerse after Keri (seminal emissions?! This suffices for conversion.)

46b (Rabah): A case occurred in which a convert circumcised but did not immerse. R. Chiya bar Rebbi, R. Oshaya b'Ribi and R. Oshaya bar R. Chiya told him 'wait, and we will immerse you tomorrow.'

We learn three laws from this:

Conversion requires three judges;

One is not a convert until he circumcises and immerses;

We do not immerse converts at night.

Question: Why don't we also learn that expert judges are required?

Answer: Perhaps it just happened that expert judges were there.

(R. Yochanan): We need three judges for conversion, for it is called 'Mishpat'

A man told R. Yehudah that he converted without Beis Din. He did not have witnesses to prove this.

R. Yehudah: You are believed to disqualify yourself, but not to disqualify your children.

Kidushin 62b: A Nochri is not able to convert whenever he wants. Conversion requires a Beis Din of three. Perhaps he will not find three to convert him.

Rishonim

Tosfos (46b DH Mishpat): We learn from "U'Shfatetem Tzedek...u'Vein Gero" (Devarim 1:16) or "U'Mishpat Echad Yihyeh Lachem vela'Ger" (Bamidbar 15:16).

The Rif and Rosh (4:31) bring the Gemara verbatim.

Question: R. Yochanan taught that three judges are needed for conversion (and the Halachah follows him! Why do the Amora'im say that Tevilas Nidah or Keri suffices?) Even according to the opinion that one judge suffices for monetary cases, here three are required, like cases of theft and hitting. Also, it is not normal for women to go with men to immerse, and women cannot judge since they cannot testify!

Answer #1 (Rif): L'Chatchilah we do not treat him like a convert and let him marry a Bas Yisrael until he immerses in front of three judges. B'Di'eved we do not disqualify the child if there were not three judges.

Bach (EH 268 DH u'Mah she'Chosav ul'Rav): The Rif holds that the Tevilah for Keri cannot make conversion even if it was in front of judges. Rather, it shows that he previously converted.

Rebuttal (Taz 8): There is no such connotation. Rashi, the Ran, the Tur and Shulchan Aruch all hold that no other immersion is necessary.

Answer #2 Rosh (ibid.): Three judges are needed only for acceptance of the Mitzvos. Even though the Gemara says that the convert immerses in front of two Chachamim, this is only l'Chatchilah.

Answer #3 (Rosh): Some say that since everyone knows that he immersed, it is as if he immersed in front of the judges.

Question: Tevilas Nidah is at night. We do not immerse converts at night (46b)!

Answer (Rosh, citing Rabbeinu Meir): The Gemara derives three judges from 'Mishpat'. I.e. "U'Shfatetem" connotes two judges, and we add one to avoid an even number. We do not derive immersion during the day from "Mishpat", for this is only mid'Rabanan. It is only l'Chatchilah.

Rambam (Hilchos Isurei Bi'ah 13:9): If we saw a convert acting like a Yisrael, e.g. she immerses after being Nidah and separates Terumah, or he immerses after Keri and does all the Mitzvos, the Chazakah is that he or she is a proper convert. This is even if there are no witnesses about in front of whom they converted. Nevertheless, we do not let them marry into Yisrael until they bring witnesses or immerse in front of us, since they were Muchzakim to be Nochrim.

Question (ha'Ramach): Why did both Amora'im prove that the conversion was valid by the fact that he or she immersed, as opposed to any other Mitzvah?

Answer (Magid Mishneh): The Rambam explains that the Amora'im were Machshir even children born previously. Perhaps it is not dependent specifically on Tevilah; they meant that since we see that they fulfill all Mitzvos like Yisraelim, surely we consider them Yisraelim. Alternatively, we must know that they immersed, since this is Me'akev conversion. The Ramban says that one who converted by himself is a total Nochri. However, if he accepted in front of Beis Din to be circumcised and immerse and they told him about some Mitzvos, even if his Milah and immersion were not in front of Beis Din he is a convert.

Poskim

Shulchan Aruch (YD 268:3): All matters of conversion, i.e. informing him about Mitzvos for him to accept them, Milah and Tevilah must be in front of three judges and during the day. However, this is l'Chatchilah. B'Di'eved, even if he immersed in front of two judges or at night, even if did not immerse for the sake of conversion, rather due to Keri or Nidah, he is a convert and may marry a Yisraelis.

Taz (8): We must know clearly that he immersed. One is not believed to say that he converted by himself (Sa'if 11) when we did not see that he immersed.

Shach (11): Tevilah in front of one suffices. However, if it was known that he was a Nochri, two witnesses are needed to prove that he immersed. This is required even when a woman converts.

Shulchan Aruch (ibid.): However, acceptance of Mitzvos in front of three and during the day is Me'akev.

Source (Gra 16): One who says that he converted by himself is believed to disqualify himself (47a). Conversion is not considered b'Yado because it requires a Beis Din of three (Kidushin 62b).

Shulchan Aruch (ibid.): The Rif and Rambam hold that if Milah and Tevilah were in front of two or at night he may not marry a Yisraelis. If he did and they have a child, we do not disqualify the child.

Rebuttal (Gra 17): The Tur says that the Rif says so, but the Rif did not say so explicitly. The Rambam is Machshir at night b'Di'eved. It seems that the Rif agrees, for he did not ask (why Tevilah Nidah, which is at night, suffices). It seems that the Shulchan Aruch only meant to cite the Rif and Rambam regarding two judges.

Beis Yosef (DH v'Chosav Od): The Rashba says that Tevilah at night is Kosher b'Di'eved because it is like the final verdict.

Shach (9): Some say that even if we began judging a monetary case at night, b'Di'eved the verdict is valid (CM 5:2)! Why doesn't the Shulchan Aruch say that some Machshir acceptance of Mitzvos at night? It seems that only regarding monetary cases the Shulchan Aruch was concerned for this opinion, because Hefker of Beis Din is Hefker.