What is an archetype? An archetype is the quintessential version of a certain person or thing, meaning that it can be described as a typical example.
In Closer, each character behaves like a different archetype despite what they all seem to have in common: an obsession with love. Alice behaves much like a classic damsel in distress on the surface, but in reality she is much more than that. Dan is a classic tragic hero, who allows his selfishness and hubris to define his actions. Larry is the antagonist, or villain, depending on how you interpret his actions, but villain and antagonist are not the same thing. A villain is someone who purposefully obstructs the protagonist's journey, and an antagonist stands in the way, but is perhaps well-intentioned or unwitting. Finally, Anna behaves much like the Madonna archetype, the perfect woman that is placed on a pedestal for all to admire. She stands strong underneath the male gaze, though her motivations can bewilder her male counterparts.
A word of caution: Do not confuse the movie starring Julia Roberts and Natalie Portman with the play as the two are actually somewhat different. For example, there is a scene between Anna and Alice wherein Alice asks for her photo negatives from Anna that was omitted from the film. Furthermore, the Anna character in the play is not American.