Kiddushin, Chapter Four, Mishnah Four
Before marrying, men were supposed to make background checks of their potential wives to uncover any potential flaws in their lineage. Our mishnah teaches that there are eight mothers whom he must check; if these eight mothers are fit to marry priests, then the man may assume that the woman whom he wants to marry is fit as well.
1) He who marries a the daughter of a priest must investigate her lineage up to four mothers, which are eight: her mother and her mothers mother, her mothers fathers mother and her mother, her fathers mother and her mother, her fathers fathers mother and her mother.
2) [In the case of] the daughter of a Levite or an Israelite, one more is added.
Section one: The first clause deals with priests who wish to marry the daughters of priests. In ancient times it was common for priestly families to intermarry with one another.
The four mothers are 1) her mother; 2) her mothers fathers mother; 3) her fathers mother; 4) her fathers fathers mother. For all four of these mothers he must also check her mother.
The Talmud explains that the mothers are checked and not the fathers since if there was a flaw in the male lineage, the men, when they argued with each other, would have cursed each other based on their faulty lineage. However, women dont argue in such a manner and therefore if there were flaws in their lineage they would not have been made known. Therefore, he must check the mothers and not the fathers.
The man must check and not the woman because women who are fit to marry priests are allowed to marry men disqualified from the priesthood, whereas priests are prohibited from marrying women disqualified from marrying priests.
Section two: Those men who wish to marry women of Levitical or Israelite families must check one more mother. It is unclear exactly what one more mother means. It either means that he goes back one more generation for each mother already checked (bringing us to 12 mothers), or that he checks one more mother on each side (for a total of 10). The reason that he must check back further is that it is more common for there to be faulty lineage in Levitical and Israelite families than it is in priestly families.