The Torah in Parashat Ki-Teseh (Debarim 23:3) commands that a Mamzer may not “enter the congregation of God.” The term “Mamzer” refers to the product of an incestuous or adulterous relationship, and the command that such a person may not “enter the congregation of God” means that he may not marry somebody from Am Yisrael. He is not actually barred from the “congregation”; he is certainly allowed to participate in all affairs in the Jewish community, and he is restricted only from marrying. In fact, the Talmud teaches that a Mamzer who is a Torah scholar takes precedence for an Aliya to the Torah over a Kohen who is ignorant in Torah.
The Sefer Ha’hinuch explains that the Torah forbade a Mamzer from marrying into the Jewish Nation because of the sinful action from which he originated. There is no doubt, the Sefer Ha’hinuch writes, that a person’s nature impacts his offspring, and thus the sinful origins of a Mamzer affect his nature and his soul. Hashem seeks to protect us from impurity and spiritual harm, and so He commanded that a Mamzer may not marry into the Jewish Nation.
Not all forbidden unions result in the status of a Mamzer. For example, if a woman had relations when she was a Nidda, then the child is not a Mamzer.
An illicit relationship confers the Mamzer status upon the product regardless of whether the sin was committed intentionally or mistakenly (such as if the man and woman thought this union was permissible).
If a non-Jewish man has relations with a Jewish woman, the child is Jewish, and is not considered a Mamzer, even if the woman was married.
If a non-Jewish man has relations with a Mamzeret (female Mamzer), then the child is a Mamzer, like the mother. If a Mamzer had relations with a non-Jewish woman, then the child is a non-Jew, and if the child converts, then he is considered a full-fledged convert, and does not have the status of Mamzer.
A Mamzer is permitted to marry a Mamzeret, and their children are considered Mamzerim. The way to terminate the “chain” of Mamzerut is for a Mamzer to marry a Shifha (gentile maidservant). Such a marriage is permissible, and the children born from this marriage have the status of servants, like the mother. Once they are released, they become full-fledged converts, and are not considered Mamzerim.
A Mamzer is allowed to marry a convert. If two converts marry each other, their children may marry Mamzerim.
This prohibition applies in all times and places. A Mamzer who married a Jewish girl, or a Mamzeret who married a Jewish man, and consummated the marriage, is liable to Malkut. The Rambam maintains that this prohibition is violated only in the framework of marriage; if a Mamzer has relations with somebody whom he had not married, then this prohibition is not transgressed. Others, however, disagree, and maintain that even relations outside the framework of marriage violate this command.