When it comes to who is to essentially to blame for the tragic ending of Romeo and Juliet, for the most part, we can agree with Prince Escalus's opinion expressed in the final scene. Prince Escalus very blatantly and accurately lays all blame on Lords Capulet and Montague:
See what a scourge is laid upon your hate,
That heaven finds means to kill your joys with love!
And I, for winking at you, discords took.
Have lost a brace of kinsmen. (V.iii.303-06)
Another important point concerning the hatred and feud can be found in the opening prologue. Shakespeare makes a point of stating that, while the feud between the two families is a longstanding one, a period of peace had been reigning, but the present generation of Montagues and Capulets had decided to rehash the old argument, as we see in the line, "From ancient grudge break to new mutiny" (First Prologue, 3). Hence it is essentially Lords Capulet and Montague's decisions that created all of the hatred and violence, leading to the play's tragic end. Had this hatred not existed, the couple's hasty marriage would not have been an issue; nor would fate have played a role in their deaths; nor would Tybalt have become so enraged by Romeo's presence at the Capulet ball, causing his own death as well Romeo's banishment and eventually Romeo's and Juliet's own deaths.
While the hatred is ultimately to blame, we must also remember that, as a tragedy, the play also contains a tragic hero. According to Aristotle's commonly accepted definition, a tragic hero must be a generally noble person who has some character flaw that leads to his/her demise. Romeo's character flaws are his impetuousness and the fact that he allows himself to be governed by his rash, passionate, intense emotions rather than by reason. Being guided by rational thought would have prevented Romeo from crashing the ball, from killing Tybalt out of revenge, and would have even helped him to realize the moment he saw Juliet in the tomb that she could not possibly actually be both dead and rosy cheeked, all of which would have prevented the couple's death in the play.
This is one of the most complex questions of the play; presumably, it is why you have been asked to write an essay on the topic. There are many reasons for the play's tragedy, and it is accurate to say that there is no single cause that all scholars agree on, nor did Shakespeare intend for it to be interpreted as such. But here are a few factors for consideration:
Haste; many people see the play as a tragedy of haste, of acting too quickly and carelessly. Consider the two lovers' hasty marriage by Friar Lawrence, Romeo's haste in killing Tybalt ("O, I am fortune's fool!"), Capulet's insistance on moving her daughter's wedding up a day, both lovers' haste in killing themselves in the end, etc. Look through the play and you will find many, many more instances of hasty action.
Fate; there is obvious reason for blaming fate and coincidence for the play's tragic ending. "From forth the fatal loins of these two foes/A pair of star-crossed lovers take their life." Consider that Capulet gives his guestlist to an illiterate servingman (who finds Romeo of all people to read it, thus inviting him to the ball), that Friar John never gets the letter to Romeo (due to a plague outbreak), that the two lovers happen to be from feuding families, etc.
Look through the play for all instances of haste, fate, coincidence, miscommunication, hatred/wrath, etc. You will find many reasons for the protagonists' ultimate demise.
I think that the feud between the Montagues and the Capulets is to blame for the tragedy at the end.