The play offers few answers with regard to how the feud between the Montagues and the Capulets started. In the Baz Luhrmann movie version, the two lords are portrayed as competing business titans. The fact that both men are lords and high-ranking in the community could lend credence to this theory. A land dispute is also possible, and would be consistent with actions of that era, although one does not get the sense that the two are neighbors.
We are given more information regarding how long the feud has lasted. While no specific dates are mentioned, Shakespeare's description of the "ancient grudge" (line 3) in the prologue suggests that this is not a new phenomenon, but that the feud has spanned several generations. This suggests that perhaps the feud was not between Montague and Capulet at all, but rather their ancestors.
The prologue also definitely tells us that the deaths of Romeo and Juliet will bring the feud to a close. In the line, "Doth with their death bury their parents' strife," (line 8) we learn that the grief that the two families endure will cause them to ultimately end the fighting.