Gemara

(Shmuel): In Avelus, the Halachah always follows the lenient opinion.

Mo'ed Katan 23a (Beraisa): An Avel within Sheloshim may not marry;

If his wife died, he may not remarry until three festivals pass;

R. Yehudah forbids the first two festivals, and permits the third.

If he does not have children, he may marry immediately, due to Bitul of Peru u'Rvu. If his wife left him with young children, he may remarry immediately, to raise them.

A case occurred in which the wife of Yosef ha'Kohen died, and in the cemetery, he told her sister 'go raise your sister's children.' Even so, he did not have relations with her until a long time.

(Rav Papa): 'A long time' means after 30 days.

Yevamos 41a (Mishnah - R. Yosi): Any woman (a widow or divorcee, from Eirusin or Nisu'in) may become Mekudeshes, but a widow must wait until the Avelus over her husband ends.

43b (Beraisa - R. Yosi): A widow must wait 30 days, due to Avelus.

Rishonim

Rif and Rosh (Mo'ed Katan 14b and 3:48): The Yerushalmi (in Yevamos) says that Avelus is 30 days for women. For men, it is three festivals if he has children, and does not have young children. If not, he is permitted immediately, like the case of R. Tarfon (it is like the case of Yosef ha'Kohen). Even though he had Nisu'in, they did not have relations for 30 days.

Rif: If one had enough children to fulfill Peru u'Rvu, i.e. a boy and a girl, he has no Heter to marry immediately for Peru u'Rvu.

Rosh: The Ra'avad obligates waiting 30 days only when one marries for the sake of young children. If one marries for children, in order to fulfill Peru u'Rvu, he may have Bi'ah immediately. If not, why may he marry her immediately? Also if one marries because he needs someone to serve him, Bi'ah is forbidden for 30 days. Also R. Tam says so. He added that the Heter to marry immediately for the sake of young children is after Shivah. At the cemetery, Yosef ha'Kohen told his sister-in-law to raise his children. This was Shiduchin (arranging a match). He did not have Nisu'in until after seven days. It is clear from Kesuvos 3b that Nisu'in is forbidden within Shivah.

Rosh (ibid.): The Ritzva says that telling her at the cemetery was Chupah. This is the episode in Zevachim 100a, in which Yosef ha'Kohen's wife died on Erev Pesach. Even if the Mes died shortly before Yom Tov, Yom Tov nullifies Shivah, so it was like after Shivah, so Nisu'in was permitted. Maseches Semachos (7:15) permits Nisu'in after 30 days if he has no children or has young children. Here we permit within Sheloshim! Rather, Maseches Semachos discusses full Nisu'in with Bi'ah. Bi'ah is forbidden within Sheloshim. R. Tam and the Ra'avad permit Bi'ah within Sheloshim for one without children. They must say that Maseches Semachos teaches that after 30 days is always permitted, but indeed for one without children, it is permitted even within Sheloshim, for Peru u'Rvu. This is just like we permit Chupah within Sheloshim, even though this is the Simchah. All we permitted applies also to an Avel over a parent. Sheloshim is the same for all Mesim.

Rambam (Hilchos Evel 6:5): An Avel within Sheloshim may not marry. One may be Mekadesh even on the day of death. If one's wife died, if he already fulfilled Peru u'Rvu, and he has someone to serve him and he does not have young children, he may not remarry until three festivals pass. If he did not yet fulfill Peru u'Rvu, or he has young children, or he has no one to serve him, he may be Mekadesh and have Nisu'in immediately, but Bi'ah is forbidden until after 30 days.

Bach (YD 392:6 DH u'Mah): Even though he cannot have Bi'ah until 30 days, we permit Nisu'in within Shivah, since it is for the sake of the great Mitzvah of Peru u'Rvu.

Rambam (ibid.): Similarly, an Avelah may not have Bi'ah until 30 days.

Beis Yosef (DH Kasav ha'Rambam): The Ramban infers that the Rambam permits an Avelah to have Nisu'in immediately. He disagrees; even Kidushin is forbidden for 30 days. I say that the Rambam holds that since we permit Kidushin on the day of death, this shows that the Halachah does not follow R. Yosi. Alternatively, we hold like R. Yosi, but one with young children may marry (immediately), like the case of Yosef ha'Kohen. However, it seems that we discuss a woman who lost her husband. For other relatives, just like a man may have Bi'ah after seven days, also a woman. A man who may remarry within Sheloshim, because he has young children or no children or no one to serve him, may not have Bi'ah until 30 days. The same applies to a woman who remarried within Sheloshim. However, the Rambam forbids Bi'ah for three months, for Havchanah (to avoid a Safek about who is the father)! It seems that he teaches that if the only concern were Avelus, she would be permitted after 30 days. He relied on what he wrote in Hilchos Gerushin (11:20) that she must wait 30 days.

Rebuttal (Bach 392:4 DH u'Mah): The Rambam would not write something that could lead people to transgress! Rather, he discusses an Avelah over other relatives (not a husband). Normally, Bi'ah is permitted after seven days. We are more stringent because she married during Shivah.

R. Akiva Eiger (on Shach 2): We can say that she married after her husband died, b'Heter. If her sister died, she may marry her sister's husband if he has young children (Rema Sa'if 3). We can say that he conducted Avelus before Yom Tov, so Sheloshim is Batel for him, and she found out only after Yom Tov. This is a Shemu'ah Kerovah, so Sheloshim still applies to her.

Gilyon Maharsha: We can say that he married b'Heter, e.g. he had young children or no children, or did not have anyone to serve him.

Rosh (Mo'ed Katan 3:48): The Beraisa mentions Nisu'in, but this is not because Kidushin is permitted. In Yevamos, R. Yosi forbids a widow to become Mekudeshes for 30 days, due to Avelus. Rather, the Chidush is that even Nisu'in is permitted after 30 days, even though one must wait three festivals regarding one's wife. Do not say that the Chidush is that Nisu'in is forbidden within Sheloshim. In any case an Avel within Sheloshim may not enter Beis ha'Simchah! Nisu'in is Simchah, even without a Seudah. He may not go even to another's Nisu'in.

Rosh (ibid.): R. Tam said that we might have thought to permit his own Nisu'in more than that of others, due to Peru u'Rvu. Even though we forbid Beis ha'Simchah for 12 months after losing a parent, he may marry after 30 days, even if he has old children. The Isur of Beis ha'Simchah is only when there is no Mitzvah. (His own) Nisu'in is a Mitzvah even if he already has many children. The Halachah follows R. Yehoshua (Yevamos 62b; even if one married in his youth,) one should marry in his old age. Rabbeinu Yosef proved that the Isur of Beis ha'Simchah is only for Reshus. Maseches Semachos (9:15) says that an Avel for any relative may not enter Beis ha'Simchah for 30 days, and a Avel over a parent may not enter for 12 months, unless it was a Mitzvah.

Rosh (ibid.): The Ritz Gei'us rules like R. Yehudah, who permits the third festival, for in Aveilus, the Halachah follows the lenient opinion. The Rambam rules like Rabanan. Presumably, the Halachah follows R. Yehudah, who is lenient. The Ramban agreed.

Tosfos (Mo'ed Katan 23a DH Ad): He must wait three festivals without Simchah before remarrying. Alternatively, it is in order to forget his old wife before remarrying, lest he think about her during intimacy with his new wife. Alternatively, perhaps he will mention his love for his old wife to vex his new wife. R. Yonah says that 'the Halachah follows the lenient opinion in Avelus' does not apply here, for the days of Avelus already passed.

Poskim

Shulchan Aruch (YD 392:2): If one's wife died, he may not remarry until three festivals pass.

Beis Yosef (DH u'Mah she'Chasav sheha'Ritz): The Ritz Gei'us and Ramban (Toras ha'Adam 198) say that presumably the Halachah follows R. Yehudah, for in Aveilus, the Halachah follows the lenient opinion. It seems that the Rif holds like the Rambam. Hagahos Maimoniyos says that also R. Chananel, Avi ha'Ezri, the Ritzva and Rashbam agree. The Terumas ha'Deshen (1) holds that this is primary.

Beis Yosef (DH veha'Mordechai): The Mordechai says that a woman need not wait three festivals because she is happy with any husband.

Taz (2): Three festivals is not for Avelus, rather, in order to forget his deceased wife. Through Simchah of the festivals, he forgets.

Shach (2): A woman is happy with any husband, therefore she forgets her first husband after 30 days. If she married within Sheloshim of other relatives, she may not have Bi'ah until 30 days. The Bach understands the Rambam to say so.

Shulchan Aruch (ibid.): Regarding entering Beis ha'Simchah, this is like the death of other relatives. Rosh Hashanah and Yom Kipur are not considered festivals for this law. If he did not yet fulfill Peru u'Rvu, or he has young children, or he has no one to serve him, he may be Mekadesh immediately and have Nisu'in after Shivah. Bi'ah is forbidden for 30 days, unless he did not yet fulfill Peru u'Rvu. Then, he may have Bi'ah after Shivah.

Taz (4): The Mechaber rules like the Rambam, who always permits Kidushin immediately. However, the Rema (in Sa'if 1) forbids even Kidushin within Shivah. Why didn't the Rema comment here?!

Shach (5): The Bach says that the Mechaber permits like he holds in Sa'if 1, that one may be Mekadesh on the day of death. We permit Kidushin and Nisu'in immediately after Shivah.

Rema: The same applies to other Avelus, even over a parent. If he did not yet fulfill Peru u'Rvu, he may make Nisu'in and have Bi'ah after Shivah. Some say that even if one is rich and could hire servants, if he does not have a daughter or daughter-in-law in the house to shampoo his head or other matters that he is ashamed to ask others to do, this is called one who has no one to serve him, and all similar matters.

R. Akiva Eiger: This connotes that a daughter-in-law may shampoo her father-in-law's head. Surely she may not delouse it! Chavos Ya'ir 135 is stringent even about shampooing.

Rema (ibid.): The custom spread to be lenient about marrying within three festivals. A Ba'al Nefesh should be stringent for himself.

Dagul me'Revavah: The custom is to be lenient in our lands because the Kidushin is on the day of the Chupah. If we would not permit Nisu'in, he will not be Mekadesh her, and perhaps someone else will be Mekadesh her first. We permit Kidushin on Chol ha'Mo'ed and Tish'ah b'Av due to this concern. We are not lenient about Nisu'in on Chol ha'Mo'ed or from Rosh Chodesh Av until Tish'ah b'Av. However, we are more lenient about Nisu'in within three festivals; we permit this for one who has no one to serve him and other cases. Even if he already started Shiduchin, perhaps she will retract. In the days of the Gemara, or in places where they are Mekadesh long before the Chupah, Nisu'in is forbidden until three festivals.