Before us is Mishnah 6 in the fourth chapter of Maseches Kiddushin, which deals with the laws of lineage: the status of the descendants of a chalal and of the descendants of a convert regarding their fitness to marry a kohen. The central principle emerging from the Mishnah is that lineage in these matters follows the father and not the mother.
Who is a chalal:
A chalal is the product of a union between kehunah and a Yisrael: a woman who is forbidden to marry a kohen and nevertheless married one, and the son born of that marriage is a chalal.
"Bas chalal zachar pesulah lekehunah le'olam":
The daughter of a male chalal is disqualified from marrying a kohen, and this disqualification continues forever. Wherever there is a male chalal, his daughter is disqualified from the kehunah; and if a son is born to that chalal, he too is a chalal, and his daughter is likewise disqualified, and so on through all the generations.
"Yisrael shenasa chalalah - bito kesheirah lekehunah":
A woman becomes a chalalah in two ways:
Through herself: a woman who is forbidden to marry a kohen, such as a divorcee, who married a kohen, becomes a chalalah.
Through her father: the daughter of a chalal is a chalalah.
In either of these types of chalalah, if an ordinary Yisrael married her, his daughter is fit for the kehunah and is not a chalalah, and likewise the son is not a chalal. For the status of chalalah does not follow the mother but the father: if the father is a chalal, his son and daughter are chalalim; and if the father is a Yisrael, the children are fit, even though the mother is a chalalah.
"Chalal shenasa bas Yisrael - bito pesulah lekehunah":
The Mishnah returns and emphasizes the other side of the principle: a chalal who married the daughter of a Yisrael, even though they are permitted to one another, passes his status on to his descendants - his daughter is disqualified from the kehunah and is considered a chalalah, since we follow the male.
The view of Rabbi Yehudah regarding a convert:
Rabbi Yehudah adds that just as in the law of the chalal we follow the male, so too in the law of the convert, for a female convert may not marry a kohen. These are his words: "Bas ger zachar k'vas chalal zachar" - the daughter of a male convert carries on her father's lineage. Just as with a male chalal his children are a chalal or a chalalah, so too the daughter of a male convert stands in the status of a convert and is not permitted to marry a kohen.
And in the reverse case, exactly as with the law of the chalal: a female convert who married an ordinary Yisrael, her daughter is no longer considered a convert according to Rabbi Yehudah, since her father is a Yisrael, and lineage does not follow the mother. The daughter is permitted to marry a kohen, just as the daughter of a chalalah who married a Yisrael is permitted to a kohen.
In summary: in this Mishnah we learned that in the laws of chalalah and conversion we follow the father: the daughter of a male chalal is disqualified from the kehunah forever, and likewise the son of a chalal is a chalal and his daughter is disqualified in all the generations; whereas a Yisrael who married a chalalah, his daughter is fit for the kehunah. Rabbi Yehudah compares to this the law of the convert: the daughter of a male convert is like the daughter of a male chalal and is disqualified from marrying a kohen, whereas the daughter of a female convert who married a Yisrael is fit for the kehunah.