Shakespeare's play, Ovidian in nature, is a play of contrasts. As such, conflict works as the counterpoint to love in this tragedy. Just as in Ovid's Metamorphoses, "violent delights have violent ends." (2.2)
Romeo and Juliet's love develops from terrible conflict: In the evening of the street fight, Romeo meets his enemy's daughter Juliet and falls in love; theirs is a violent love, a "violent delight."
There is constant conflict: When Romeo comes to Friar Laurence, he does not wish to join the two lovers in marriage; however, Romeo's behavior convinces the priest that it would be better if he and Juliet were married. Romeo feels great delight to be married to Juliet, be he soon comes into great conflict with Tybalt, who yet holds "the ancient grudge,"; they fight and Romeo slays Tybalt. Then, because he has violated the command of the Prince of Verona, Romeo is banished from the city. Learning of his banishment and his murder of Tybalt, Juliet's love for her cousin and for her husband now conflict.
In addition, she comes into conflict with her parents who insist that she marry Paris, a wealthy aristocrat, so that she will recover from the loss of her beloved cousin Tybalt. Unbeknown to her parents, however, is the fact that she is married already, so in her violent personal conflict, Juliet finds herself in a quandary. In order to deal with this conflict, she follows the directions of Friar Laurence and ingests a vial of a sleeping potion, which simulates death for a short time. During this time that Juliet is interred, the friar hopes to ameliorate the relationship between Montagues and Capulets.
Malevolently, Fate enters in as Friar Laurence's messenger is turned away in Mantua where he has come into conflict with authorities there who forbid him to enter the city because of a plague. In the meantime, Romeo hears from his servant that Juliet is entombed in Verona. Amid great inner conflict as his emotions are desperate, Romeo rushes to an apothecary, buys poison, then he heads to Verona in order to find Juliet. Believing that she lies dead in the family tomb, he enters, and kills Paris who attempts to enter the catacomb. Finding Juliet limp and pale, Romeo kills himself. Not long afterwards, Juliet is revived from the potion. Tragically, however, she discovers the body of Romeo, and she, too, commits suicide and also dies.
In the first act, Romeo has had a premonition of all that is to come; he senses the conflict, the acts of fate:
I fear, too, early. For my mind misgives
Some consequence, yet hanging in the stars,
Shall bitterly begin his fearful date. (1.4)
In the end, the Prince declares, "Some shall be pardoned, and some punished." (5.3)
Think of it this way: any main problem within a story can be called its conflict. In Romeo and Juliet, the feud between the Montagues and the Capulets serves as a primary conflict, and its results are fatal. See "tdot teacher's" reply for more specific types of conflict: Character vs. character, character vs. self, character vs. nature, etc.
While other conflicts are present in this story, the family feud is its central pivoting point of contention. Without the dispute between the families, the whole story falls apart. Such is the case with any main conflict in literature; without it, no story would be complete.
Conflict is defined as the basic opposition of two forces that serves as the basis for literature. It is generally divided into categories such as Human versus Self, Human versus Society, Human versus Human, Human versus the Supernatural. In this play, we have examples of all these types of conflict: Human versus Self (Romeo and his feelings for Rosaline; Juliet and her duty to her parents), Human versus Society (the feud), Human versus Human (Romeo and Tybalt; Capulet and Montegue), Human versus the Supernatural (the role of Fate in the play).