Gemara

(Mishnah): If one wants to marry a Bas Kohen, he must check the lineage of four of her maternal ancestors, and also their mothers, making eight in all:

Her mother; the mother of the father of her mother; her father's mother; the mother of her father's father; and the mothers of these four women.

To marry a Bas Levi or Bas Yisrael, he must check these and another mother.

Question: Why do we check only mothers, and not fathers?

Answer: When women quarrel, they insult each other about immorality. Lineage problems are not exposed. When men quarrel, they insult each other with lineage problems. Any problems would be common knowledge.

Question: Why don't women need to check a man before marrying him?

Answer: Rav taught that women are not forbidden to marry Pesulim.

(Rav Ada bar Ahavah): Checking the four mothers entails checking 12 women;

(Beraisa): Checking the four mothers entails checking 16 women.

Question: Granted, Rav Ada discusses a Bas Levi or Bas Yisrael. We add one mother to each of the four (her grandmother). How does the Beraisa find 16?

Answer: The Mishnah says 'we add an additional mother', i.e. an additional pair of mothers (her grandmother and great grandmother).

(Rav Yehudah citing Rav): Our Mishnah is like R. Meir, but Chachamim say that all families are assumed to be of proper lineage.

Version #1 - Question: Rav Chama bar Gurya cited Rav to say that the Mishnah is only when there is Ir'ur (protest)!

Answer: The Amora'im argue about Rav's opinion.

Version #2 (Rav Chama bar Gurya citing Rav): there is Ir'ur, we must check her ancestors (even according to Chachamim).

Rishonim

Rambam (Hilchos Isurei Bi'ah 19:18): If there is Ir'ur against a family, i.e. two testified that a Mamzer or Chalal was mixed in, it is a Safek. If the family is Kohanim, to marry a woman from it one must check four mothers, who are eight in all: her mother; the mother of the father of her mother; her father's mother; the mother of her father's father; and the mothers of these four women.

Ran (31b DH Tanan): Rashi says that two witnesses put out a rumor that the lineage is tainted. The Rambam says that they testify that a Pasul was mixed in. This is only for Pesulei Kahal, for Yisre'elim recognize the Mamzerim among them, but we are always concerned for Chalalim. Other say that we are not concerned even for Chalalim unless there is Ir'ur.

Rambam (19): If it is a family of Leviyim or Yisraelim, he must check another pair, 10 in all. This is because more Pesulim are mixed among Leviyim and Yisre'elim.

Question (Magid Mishneh): The Rambam explains like Rashi, that any man must check more when marrying a Leviyah or Yisraelis. However, no opinion in the Gemara requires checking less than 12! This requires investigation.

Answer (Gra EH 2:16): The Rambam's text of the Beraisa said that one must check 10 for a Leviyah or Yisraelis. The Gemara answered that one must check one more pair in all, not another pair above each mother.

Prishah (18): He checks one more from her mother's side, and one more from her father's side. However, we have no source saying which is added! Perhaps we check an additional mother of the maternal grandmother on each side, lest the Bedikah extend too many generations into the past.

Etzei Arazim (in Otzar ha'Poskim): He checks her totally maternal great grandmother and great great grandmother, which parallel her father's maternal grandmother and the maternal grandmothers of her grandfathers.

Rambam (21): A woman who wants to marry into such a family need not check, because women are not forbidden to marry Pesulim.

Ran (ibid.): Since the Torah did not forbid women to marry Pesulim, also Chachamim did not require them to check. Some texts require a Kohen who marries a Kohenes to check four mothers for Pesulei Kehunah. This Bedikah is only for four. Anyone, even a Kohen, must check another one for Pesulei Kahal. Also a woman must check a man for Pesulei Kahal, but she need not check for Pesulei Kehunah. The Yerushalmi says that the entire Bedikah is a fine, so that one will marry within his family, about which he already knows. Chachamim were more stringent about Leviyim and Yisre'elim, who are not so exacting about lineage, or because there are more Pesulim among them.

Poskim

Shulchan Aruch (EH 2:3): If there is Ir'ur against a family, i.e. two testified that a Mamzer or Chalal or Avdus (slaves) became mixed in, it is a Safek. If the family is of Kohanim, to marry a woman from it one must check four mothers, who are eight in all: her mother; the mother of the father of her mother; her father's mother; the mother of her father's father; and the mothers of these four women.

Tur: Perhaps even though Chachamim do not require Bedikah, we follow the Stam Mishnah, which does

Rebuttal (Beis Yosef DH u'Mah she'Chosav Rabeinu d'Ifshar): Rav taught that Chachamim argue with the Mishnah, i.e. the Halachah does not follow the Mishnah. This is why the Rif and Rosh omitted the Mishnah. Why didn't they bring it to teach about when there is Ir'ur? Perhaps they never require the Bedikah of the Mishnah. When there is Ir'ur; any Bedikah suffices.

Chelkas Mechokek (2): The Shulchan Aruch is concerned for Avdus, i.e. also Shifchos were mixed in. We would not be concerned for only male slaves, for their children are Kesherim. The Rambam is Machshir even to Kehunah.

Beis Shmuel (8): We need not check men, for their Pesulim become known. If so, why must we check her father's mother? We must say that we are concerned lest his mother was a Mamzeres. This would not become known.

Chazon Ish (EH 1:20): This is not incomprehensible. If so, we should need to check also him! It seems that the Gemara means that when men quarrel, they insult each other with lineage problems, so if any problem was known, it would be commonly known (without investigation). A known Pesul of a woman could be stifled, so we must investigate women. Checking is finding that she was Muchzekes to be Kesherah where she was born, or perhaps, any place she lived.

Shulchan Aruch (ibid.): If it is a family of Leviyim Yisraelim, he must check another one, 10 in all. A woman who wants to marry into such a family need not check, because women are not forbidden to marry Pesulim.

Chelkas Mechokek (3): The Magid Mishneh holds that if the Ir'ur is about Chalalim, only a Kohen must check. If it is about Pesulei Kahal, even a Yisrael must check. We require checking less if her family is of Kohanim, even if the Ir'ur is about Pesulei Kahal.

Question (Chelkas Mechokek 5): The Rambam says that Ir'ur is testimony that a Pesul was mixed in. If so, we should require even a woman to check!

Answer #1 (Beis Shmuel 9): Perhaps we are lenient because everyone in the family is a Safek Mamzer, and mid'Oraisa he is Kosher. Rashi said that the Ir'ur was rumors, for if there was real testimony, she would need to check. The Ba'al ha'Ma'or and Milchamos Hash-m hold like the Ran; they require her to check whenever there is Ir'ur of Mamzerus. One should be stringent.

Answer #2 (Taz 2): The Rambam holds that women are not commanded, even when there is Chezkas Isur. Even though Leviyim and Yisre'elim are not commanded about Chalalos, they have no special leniency. Women have a special leniency: Kohanos are exempt from an Isur that applies to Kohanim.