In my opinion, the major conflict in Romeo and Juliet is the feud between the Montagues and the Capulets. Evidence for this theory is given in the prologue to the play.
"Two households, both alike in dignity
(In fair Verona, where we lay our scene),
From ancient grudge break to new mutiny,
Where civil blood makes civil hands unclean.
From forth the fatal loins of these two foes
A pair of star-crossed lovers take their life,
Whose misadventured piteous overthrows
Doth with their death bury their parents' strife.
The fearful passage of their death-marked love
And the continuance of their parents' rage,
Which, but their children’s end, naught could remove."
The conflict is described as ancient, and the play opens with a new mutiny born of this ancient grudge. The prologue suggests that their love is doomed from the start by Romeo and Juliet's parents' anger, which nothing could stop, except for the deaths of their children.
The events of the play begin with a brawl in the town center between servants of the house of Montague and Capulet. The prince threatens both families with the punishment of death if the streets are disturbed in this way again.
The conflict between Tybalt and Romeo comes from the feud between these two families. Tybalt is insulted that Romeo and his friends come uninvited to a party thrown by the Capulets. He entices Romeo to fight, but Romeo refuses. Romeo's quick-tempered friend Mercutio doesn't resist fighting, and Mercutio is tragically killed when Romeo tries to stop the fight.
In response to his friend's death, Romeo seeks revenge and kills Tybalt. This results in his banishment from Verona. Romeo and Juliet have already secretly married at this point, so the friar concocts a plan to fake Juliet's death so that she can escape with Romeo because the two are inconsolable. This plan would have worked, but the communication the friar sends to Romeo never reaches him. When Romeo finds Juliet dead, he kills himself. She awakens to find him dead and kills herself. They are so desperate to be together that they can't imagine life without each other. They can't tell anyone in their family about their love because of the feud between the families.
In the end, through the death of the two young lovers, their parents are able to make peace. But this peace comes at a tragic price. At the root, all the deaths in this play are a result of the ancient conflict between the Capulets and the Montagues.