The kohanim are the descendants of Aaron, chosen by G-d at the time of the incident with the Golden Calf to perform certain sacred work, particularly in connection with the animal sacrifices and the rituals related to the Temple. After the destruction of the Temple, the role of the kohanim diminished significantly in favor of the rabbis; however, we continue to keep track of kohein lineage.
Kohanim are given the first aliyah on the Sabbath (i.e., the first opportunity to recite a blessing over the Torah reading), which is considered an honor. They are also required to recite a blessing over the congregation at certain times of the year.
The term Kohein is the source of the common Jewish surname Cohen, but not all Cohens are Koheins and not all Koheins are Cohens.
Sources: Judaism 101