What does it mean to call someone a "Romeo"?

In Shakespeare's Romeo and Juliet, Romeo is seen at the beginning of the play head over heels in love for Rosaline. Although Romeo is infatuated with her, Rosaline doesn't seem to share these feelings, as Romeo notes: "She hath forsworn to love, and in that vow / Do I live dead that live to tell it now" (1.1.223–224). Once he sees Juliet (on the same day, no less), he falls immediately head over heels in love again: "Did my heart love till now? Forswear it, sight! / For I ne'er saw true beauty till this night" (1.1.50–51).
It is this speedy transition that likely gives Romeo a negative reputation in popular culture. Although the character in the play does seem to truly love Juliet, his speedy transition from one girl to the next is at least a little questionable. Did he really love Rosaline at all? Or did she just represent his quest for the unattainable? Also, anyone who doesn't believe in love at first sight will certainly be a bit skeptical about Romeo's immediate profession of true love over the sight of Juliet.
So, in our society, a Romeo is a type of guy who says what he thinks girls want to hear in order to win them over. He is fickle and not subject to settling down or committing to just one girl. Some of these characteristics can be attributed to the fickle ways of Shakespeare's Romeo, but the play's character did commit to Juliet, marrying her within about twenty-four hours. It's a bit of a misnomer.

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.

Shakespeare's Romeo and Juliet has become a modern symbol for archetypical young love. The terms "Romeo" and "Juliet" (particularly the less common name "Romeo") have become synonymous with "lover." That's how ubiquitous the play has become! If you say to anyone, "He's such a Romeo," they'll know exactly what you mean.