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A Cohen is one of the class of priests among the Jewish people. Membership is hereditary, via the father. A slew of common Jewish surnames (Cohen, Kagan, Kahane etc) are variations of Cohen and usually indicate that the holders are Cohanim.
Cohen simply means priest in biblical Hebrew (as well as in Canaanite and Ugaritic). However, Chief Rabbi Sacks has argued persuasively that the primary meaning of Cohen was teacher. Just before the Torah is given, God promises that Israel will
be a kingdom of priests (Exodus 19:6). This is an aspiration to universal literacy according to Rabbi Sacks; in the ancient world, only the priests could read.
The priests performed their full range of functions when the Temple stood. Today, the rights and duties of Cohanim include being called up for the first aliyah to the Torah and giving the community the priestly blessing in synagogue ("May the Lord bless you and keep you" etc, Numbers 6: 24-25).
The often heard objection, ("Why does Joe Cohen get to bless me; he's certainly not so holy") relies on a misunderstanding. The Cohanim are servants of the people. This insight is preserved in modern Hebrew, where the verb l'cahen means to serve in any public capacity.
The Cohanim bless the people "with love." The power of invoking blessing rests with anyone who feels love and concern for the object of their blessing, for example parents who traditionally give their children the priestly blessing on Friday night.