Obviously, the central conflict in Romeo and Juliet provides the backdrop for the whole story. It is, of course, the blood feud between the Capulets and Montagues, We do not learn the cause of the feud, which apparently dates back to before any of the characters can remember. But it has divided the whole city, and ultimately claims the lives of Tybalt, Mercutio, and in a sense, Romeo and Juliet themselves. In the context of the play, this feud makes the play a conflict between love and hate.
Another, more thematic conflict is that between free will (or to put it another way, human agency) and fate. Romeo and Juliet are described as "star-cross'd lovers" in the Prologue, but it is unclear whether Shakespeare wants us to view their fates as a matter of destiny or the product of their decisions and the decisions of those around them, particularly the Nurse and Friar Lawrence.
Other conflicts in the play might to said to include youth against old age, haste against caution, and loyalty to family against pursuing one's own path.