My last name is Cohen, and I was told that Jews with this name have a special designation. Is this true?
According to the Torah, one is a Kohen, a Levite or an Israelite. Kohanim (plural of Kohen) and Levites are descendants of Aaron, the High Priest. Much of the work of the priests involved sacrifices made at the Temple. As a result, the Kohanim and Levites had certain privileges and certain restrictions. Among the restrictions was that Kohanim could not come into contact with a corpse unless it was of a close relative, lest they be rendered impure. Today, some Jews whose last name is Cohen, Kahn, or something similar and who believe they are descendants of the priestly class may still observe certain restrictions by, for example, not entering cemeteries.
Reform Judaism is a practice dedicated to the religious equality of all Jews and does not differentiate between Jews based on ancestry. Therefore, Reform Jews - even if they believe they can trace their lineage to the priestly class - do not typically observe the practice.