The term protagonist is derived from ancient Greek and means 'player of the first part' or chief actor. A few characters in a play may, however, play equally important roles which may lead to a situation where you have not only one protagonist. Such is the situation in Romeo and Juliet. It is commonly believed that the protagonist is the 'good guy' or hero of the story, but the anti-hero or villain may just as much be the protagonist, if he or she is the chief character.
The antagonist is the character in the play who opposes the protagonist. There may obviously also be more than one antagonist. It is also important to realise that the antagonist is not necessarily a person only, but may be a situation or event that acts as a stumbling-block to the protagonist - this is also true in Romeo and Juliet.
In Romeo and Juliet, the two 'star-crossed lovers', are the obvious protagonists. They are responsible for, and central to, the unfolding of events in the play.
The antagonists, in this regard, would firstly be the 'age-old feud' between the two opposing families, the Montagues and the Capulets. Since the protagonists are each a member of an opposing family, the conflict creates a major stumbling-block in their attempt to exercise their love for one another. The feud culminates in their tragic deaths.
Because of the feud, Romeo and Juliet's parents also are protagonists. As the heads of their households, they are against the idea of the two lovers being involved in a romantic liaison. It is almost impossible for them to consider the idea of their beloved offspring being romantically involved with each other - their contempt for each other creates the barrier that impedes our protagonists' love to flourish freely and openly.
Tybalt, a loyal member of the Capulet family, is a clear protagonist to especially Romeo, whom he openly despises. As a barrier to Juliet's affair with Romeo, he also becomes an antagonist to her, since he prevents her from achieving true happiness. By killing Mercutio Romeo's best friend, he deepens and escalates the conflict between the families. Ironically, when Romeo kills Tybalt in revenge later, he performs an antagonistic act which results in his banishment, worsening the situation.
To a lesser extent, even Paris, in his desire to marry Juliet, unknowingly plays the role of antagonist since his willingness to be her husband forces Juliet's hand and she has to consider desperate measures to escape marrying him, ultimately leading to both his and her demise.
It is clear that, because of the complicated issues at play, there is more than one protagonist or antagonist in this tragic drama.