Traditional gender roles constitute a key part of the cultural background against which the action of the play unfolds. In Renaissance Verona, where Romeo and Juliet takes place, such roles are fairly rigid and unyielding. Men are men and women are women, which in practice means that men engage in traditional manly activities like fighting and defending their family honor, while their wives remain chaste, pure, and demure.

And yet both Romeo and Juliet challenge the conventional gender roles of their time as they pursue their secret relationship. For his part, Romeo displays signs of femininity in his capacity for falling head over heels at the drop of a hat. This earns him the gentle mocking of his male friends, who subscribe to the dominant cultural understanding of what a man should be and how he should behave.

As for Juliet, she goes against the prevailing norms by defying her father. Lord Capulet, in keeping with long-standing convention, has decided that his daughter shall marry Paris. Having made that decision, he expects Juliet to obey his wishes without question. But Juliet hasn't gotten the memo: she refuses to go along with what her father wants. She chooses instead to follow the dictates of her heart, something that was exceedingly rare among aristocrats of the time.

As she doesn't love Paris, she refuses to get married to him, earning her a furious rebuke from her father. Traditional gender roles dictate clearly that Juliet should do what her father tells her to do. But in opting to marry Romeo, the man she loves, Juliet is mounting a serious challenge to those gender roles and what they represent.

How are masculinity and femininity represented in Romeo and Juliet?

I would agree with the other answers. Shakespeare, in his typical way, both defies and upholds gender norms. He defies gender norms by giving Juliet a strong, self-willed character, almost as impulsive as Romeo and certainly not submissive to patriarchal authority. At the same time, Juliet's role in the play is more passive and a bit more cautious than that of Romeo and his friends. It is Romeo who scales the walls of her home, at great risk to himself, to call out to her on the balcony; in this instance, it is Juliet who urges caution. On their wedding night, it is again Juliet urging Romeo to leave before he is discovered. Juliet also does not run around provoking her enemies. 

Instead, it is the males like Mercutio, Tybalt, and Romeo who occupy the public spaces and get into sword fights. It would be difficult to imagine, say, Juliet and Lady Montague engaged in a sword fight on the streets of Verona.

Juliet may speak first of marriage, but she would know very well that the stakes of sexual passion are higher for her: she cannot afford to become a fallen woman. Romeo, we learn early on in the play, is not terribly concerned about the order of sex and marriage. He cannot understand why Rosaline does not share his desire to jump into bed. He and his friends tend to see woman primarily as sex objects: they are on the prowl for beautiful women, not intellectual companions. 

Men use sex talk to intimidate woman, as Mercutio tries to do with Juliet's nurse. He does not manage to intimidate her, but he does manage to offend her. 

Masculinity and femininity are portrayed quite differently in Romeo and Juliet by William Shakespeare, as gender roles were very distinct in his period. 

The first difference in gender roles is that physical violence, in the form of fighting, rioting, and dueling, is portrayed as the exclusive dominion of males in the play. The only physically violent act committed in the play by a woman is Juliet's suicide, and that is violence towards herself not towards others. 

Next, raising children is the task of women, whether mothers in the lower classes or hired nurses in the upper classes. Thus the Nurse functions as a mother figure in the play and Juliet's father is a distant authority figure.

Taking initiative in relationships is also a distinctively masculine role; while Juliet can accept or reject Romeo's advances, she couldn't just walk up to a man and ask him to dance at a ball.

Finally, the priesthood, and thus religious authority, in the play is exclusively masculine. Only a male priest could marry the young lovers.