A Romeo is the kind of guy—like the "real" Romeo, from the Shakespeare play—who is in love with being in love. Chasing romance is his life's true passion. He is always falling head-over-heels in love. He then woos the current girl of his dreams with his his poetic love language, comparing her eyes to the stars and her radiant face to the sun. He does everything he can to get her to fall in love with him, just as Romeo did with Juliet. And, like Romeo, he doesn't wait around. He dives right in, because love is what he does. He lives for love.
However, the term has attained a somewhat negative connotation. This is a little unfair to the "real" Romeo, in my opinion. A "Romeo" often refers to the kind of guy who loves girls and leaves them, always moving on to the next attraction. At the same time, this characterization is not entirely unfair, because Romeo does moon around and claim undying love for Rosaline, only to completely forget about her the moment he lays eyes on Juliet. However, since he kills himself over losing Juliet, we have to believe that love was sincere.
In short, a "Romeo" is a man who is always falling in love and speaking the language of love in a way that may or may not be sincere—and perhaps making the other guys jealous with his poetic approach.