Who and what is a Cohen?

Following Jewish tradition Cohen's are descendants of Aäron, the first high priest in Judaism and Mozes' older brother. All variations of the name Cohen, like Cahen, Kohn, Kahn, Kahane, Cohn and Coen are Cohen's (also known as kohanim) and play an important part in the Jewish worship. The 'priesthood' only applies under strict circumstances: no ancestors may have been married to a divorced, converted or non-Jewish woman. Also, the Cohen in question must be a direct descendant of Aäron (as direct as possible through the millennia). This seems quite hard to prove, even though recent DNA-research suggests that ninety percent of Cohen's nowadays are genetically linked to ancestor Aäron. However, the Cohen-gen is scientifically controversial, has no halachic status, and is therefor not acknowledged by Judaism.