There are certain women whom a Kohen is forbidden to marry, such as a divorcee. If a Kohen marries such a woman in violation of Torah law, then his children have the status of "Halalim." A son born from such a marriage ("Halal") is not considered a Kohen, even though the father is a Kohen, and a daughter born from such a marriage ("Halala") may not marry a Kohen. The Kohen himself, however, retains his status as a full-fledged Kohen despite the fact that he violated the special code of law of Kohanim. Even though he committed a Torah violation by marrying a woman forbidden for Kohanim, he is nevertheless considered a valid and ordinary Kohen.
The Aruch Ha’shulhan (Rabbi Yechiel Michel Epstein of Nevarduk, 1829-1908) raises the question of why this should be the case. Why are the Kohen’s sons disqualified for the Kehuna (priesthood), whereas he, who committed the offense, remains a full-fledged Kohen?
The Aruch Ha’shulhan answers by noting the principle mentioned in the Gemara known as "Yisrael, Af Al Pi She’hata Yisrael Hu" – "A Jew is a Jew, even if he sins." A Jew does not forfeit his status as a Jew if he or she violates the Torah. Sinners remain full-fledged Jews, despite the violations they committed. By the same token, a Kohen retains his status as a Kohen which the Torah conferred upon him, even if he violates the laws of the Kohanim. The children, however, never had a status of Kehuna such that we could say that they retain their status despite the sin that had been committed. And therefore since they were conceived in a forbidden marriage, they do not obtain the status of Kohanim.
As mentioned, if a Kohen marries a woman forbidden for him, his sons have the status of "Halal" and are not considered Kohanim. They are allowed to marry any Jewish girl, just like ordinary Jewish men, but his children inherit his status and are considered "Halalim." Even though the marriage is entirely permissible, his children are "Halalim" just as he is a "Halal." Their last name might be "Cohen," but since they are "Halalim," they are not treated as Kohanim. This continues in the next generation, as well, meaning, the children born to the sons of a "Halal" are likewise "Halalim," and they bequeath this status to their sons, and so on.
A "Halala," however – meaning, the daughter born from a marriage between a Kohen and somebody forbidden for him – does not pass down this status to her children. If a "Halala" marries an ordinary "Yisrael" and has children, her children are considered ordinary "Yisraelim." Of course, the sons will not be Kohanim, since the father is not a Kohen. They are considered simply ordinary "Yisraelim," and the daughters are considered ordinary Jewish girls. They are not considered "Halalot," and may thus marry Kohanim, even though their mother is a "Halala."
Summary: If a Kohen marries a woman forbidden for Kohanim, he remains a full-fledged Kohen. A son born from this marriage, however, is considered a "Halal," and does not have the status of a Kohen. A daughter born from this marriage is considered a "Halala," which means that she may not marry a Kohen. The children of a "Halal" are also "Halalim," whereas the children of "Halala" are not considered "Halalim." Thus, the sons of a "Halal" are not considered Kohanim and they pass down the "Halal" status to their children, and the daughter of a "Halal" may not marry a Kohen. The children of a "Halala," by contrast, are treated in Halacha as ordinary Jews.