Masechet Kiddushin (Tractate Kiddushin), Perek Gimel (Chapter Three), Mishnah Yud Alef (Mishnah Eleven). The Mishnah says: In any situation where a couple has Kiddushin, they’re able to halachically get married, and there’s no Avera (sin) in that marriage, the status of the child goes after the father. What’s an example of this? It’s a regular Jewish marriage, a case of a girl who’s either a Kohen (priest), a Leviyah (female Levite), or a Yisraelit (Israelite woman), one of the three, and she marries a Kohen, Levi (Levite), or Yisrael (Israelite) man. Since this was a halachically acceptable marriage, Kiddushin takes effect, and the child that comes from them takes the status of the father. For example, if the father was a Kohen and the mother was a Yisraelit, the child born from them is now a Kohen.
But in a situation where there is Kiddushin between them and Kiddushin works, but the Kiddushin is asur (forbidden), the status of the child goes after the problematic component of that marriage. What’s an example of this? A widow who gets married to a Kohen Gadol (High Priest), or a divorced woman or a woman who went through Chalitza (levirate release) who gets married to a regular Kohen. In these situations, the woman becomes a Chalalah (woman disqualified from priestly status) in this marriage, and the child then acquires that blemish and becomes a Chalal (person disqualified from priestly status) as well, after the mother.
You have a Mamzeret. Mamzeret is a product of an adulterous marriage, where a man was with a woman he wasn’t supposed to be with, on the punishment of Karet (divinely imposed excision), and a Natin is a person who comes from the Giv'onim (Gibeonites). So you have a regular Jewish girl or Jewish man who married either a Mamzeret or a Netinah (female Natin), or Bat Yisrael le-mamzer u-le-natin (an Israelite woman to a mamzer and a Natin), or the regular Yisrael girl who married a Mamzeret or a Natin. In these situations, the child goes after either the mother or the father, whichever one of them is Pagum, whichever one of them is the problematic part of the relationship.
Now we’ll talk about a situation where there’s not even Kiddushin: a situation where a woman has relations with a man, but there’s never a possibility of Kiddushin between them, even though this woman would have the option of Kiddushin with another person. Ha-vlad mamzer (the child is a mamzer). The child that she has is now a Mamzer. The child that she has with that man she wasn’t allowed to be with, where there’s no possible Kiddushin, is a Mamzer. What’s the situation? This is a situation where somebody has relations with any form of an Erva (forbidden sexual relation), like a close relative, for example. A man has relations with his sister. She’s technically allowed to get married to somebody else; she’s a regular Jewish girl. She’s allowed to get married to somebody else, but there’s never any possible situation of Kiddushin between a man and his sister. So there’s no Kiddushin between them. When he has relations with her, his close relative, and they have a child, that child is a Mamzer.
A woman who has relations with a man and can’t have Kiddushin with him, can’t get married to him, and can’t have Kiddushin with anybody else either—there’s never a time when this woman can have Kiddushin. We say that the child born from this relationship is the same status as the mother. What’s an example of this? The child of a Kena'ani (Canaanite) slave woman or the child of a non-Jew. These women can’t be engaged. They can’t get married to a Jew. So, if a regular Jewish man has relations with one of them, what happens in that situation is that her status is passed on to her children. She passes on her status as a Kena'ani slave to her children. And that would tell us that if a Jewish man marries a non-Jewish woman, the children are not Jewish. But the inverse, we learn from the Pesukim (verses): if a non-Jewish man had relations with a Jewish woman, that child would be Jewish.