From Wikipedia (7 January 2022):
Cohen (Hebrew: כֹּהֵן, kōhēn, "priest") is a Jewish surname of biblical origins (see: Kohen). It is a very common Jewish surname (the most common in Israel), and the following information discusses only that origin.
Bearing the surname often (although not always) indicates that one's patrilineal ancestors were priests in the Temple of Jerusalem. A single such priest was known as a Kohen, and the hereditary caste descending from these priests is collectively known as the Kohanim. As multiple languages were acquired through the Jewish diaspora, the surname acquired dozens of variants. Not all persons with related surnames are kohanim, and not all kohanim have related surnames.
Some Kohanim have added a secondary appellation to their surname, so as to distinguish themselves from other Kohanim—such as Cohen-Scali of Morocco, who trace their lineage to Zadok, and Cohen-Maghari (Meguri) of Yemen, who trace their lineage to the first ward, Jehoiariv, in the division of twenty-four priestly wards.
Being a Kohen imposes some limitations: by Jewish law a Kohen may not marry a divorced woman, and may not marry a proselyte (someone who converted to Judaism). Nor should an observant Kohen come into contact with the dead or enter a cemetery.
An effort to trace whether people named 'Cohen' actually have a common genetic origin has been undertaken, using a genealogical DNA test associated with the Cohen Modal Haplotype.