Cohen is a very common Jewish surname. But there's more to it than that.
Within Orthodox Judaism there are three "groups" - "Cohanim" (the plural of "Cohen"), "Levi'im" (the plural of "Levi") and "Yisraelim" (the plural of Yisrael). Back in the days of the Temple, these decided what you would do.
The Cohanim were the Priests - they ran the services and performed the sacrafices. They didn't have outside jobs, as they spent their entire life working for the Temple. Therefore, they were allowed to eat some of the meat that was brought for sacrafice.
The Levi'im were the "assistants". They had various other roles in the temple, including being the musicians. They also didn't have outside jobs. Both the Cohanim and Levi'im are descended from the tribe of Levi, one of the twelve sons of Jacob. All the descendants became Levi'im, except the descendents of Aaron, Moses' older brother, who was the first Cohen. This "status" is passed down by a man to his sons.
The rest of the Jewish community are Yisraelim.
Although they had many responsibilities in the Temple, almost all of those are not done nowdays (Jews don't do animal sacracfices at the moment). The one thing left is performing a blessing on festivals (and every day in Israel), which is left to the Cohanim to do, and the Levi'im assist, like in the Temple, by washing the hands of the Cohanim beforehand. A Cohen also has the rights to be called up to the reading of the Torah first, followed by a Levi, then 1 or more Yisraelim. There are also a few other minor significances.
I think this is in some way similar to the Hindu Caste system. However, there is no concept of marrying only within your "group", and this is in no way encouraged or discouraged.
So Jews who have the name "Cohen" probably are, or at some point were, Cohanim. It's not always the case though, and there are Cohanim not called Cohen, and other Jews who are called Cohen!
I believe non-Orthodox streams of Judaism don't make the Cohen/Levi/Yisrael separation any more.
Two Sheds says Conservative Judaism also distinguishes Cohens, though I'm not sure about Levis. I'm also not sure about Reform and other non-Orthodox movements