pouring poison into his ear whilst he was sleeping in the orchard. What makes the crime even more malevolent is the fact that Claudius had seduced the virtuousGertrude, Hamlet's mother, and married her after he had killed his own brother, a most unnatural act. Furthermore, king Hamlet was not even granted the opportunity to seek divine redemption, for he was murdered in his sleep. Because of this, he was damned to everlasting perdition. All of these should urge Hamlet to a speedy vengeance, which he has promised. As dawn approaches, the ghost disappears.

Hamlet later speaks to Horatio and Marcellus and swears them to secrecy about what they had witnessed. It is then that he speaks the lines quoted in the question. By saying that 'The time is out of joint', Hamlet means that everything has turned topsy-turvy, the natural order of things has been turned upside down, nothing is normal or as it should be. These references stem from what he has just learnt from the ghost and also the current situation in Denmark, which is preparing itself for an invasion by a rebel Norwegian force led by Prince Fortinbras.

In saying 'O cursed spite', Hamlet is suggesting that the forces of destiny have been spiteful and have cursed him. Why? 'That ever I was born to set it right.' He feels cursed about his birth, not about the fact that he has to correct the wrongs in Denmark. The unnatural order has to be restored. Hamlet does not question this fact, but he does question why he was born to do it.

Hamlet has promised to adopt an antic disposition - feign madness - to enact his revenge, but we learn later that he procrastinates to such an extent that many others die as well. This is probably because he was not really that willing to do his duty, in spite of the solemn pledge he had made to his father's ghost.

Hamlet is upset because quite a lot of things seem to be going wrong.  First, his father dies.  Then, his mother remarries incredibly quickly, and, still worse, she marries Hamlet’s uncle, her own brother (by biblical precedent), making him Hamlet’s new stepfather.  This new stepfather, Claudius, allows Laertes to return to the continent, but he refuses Hamlet’s request to return to school in Wittenberg— a request which seems perfectly reasonable—all while insulting Hamlet’s masculinity.  Now, Hamlet has learned that his uncle/stepfather actually murdered his father, the old king, in cold blood.  Absolutely everything in Hamlet’s life seems out of whack—or "out of joint"—and he curses the fact that it really falls to him to avenge his father’s murder and set everything right again.  It is certainly his job to do so, both as old King Hamlet’s son as well as the person who his father’s ghost appointed.  He is now the only person, besides Claudius, to know the truth, and he must find a way to right all these wrongs done by others. 

Hamlet has just finished talking to the ghost and finding out that his father was murdered by his uncle, Claudius. The ghost has left and Hamlet swears Marcellus and Horatio to an oath not to reveal what they have seen or what Hamlet is trying to do, even if he pretends to be mad:

HAMLET
    ...
    There are more things in heaven and earth, Horatio,
    Than are dreamt of in your philosophy. ...
    How strange or odd soe'er I bear myself,
    As I perchance hereafter shall think meet
    To put an antic disposition on,
    That you, at such times seeing me, never shall,
    ...
    ... by pronouncing of some doubtful phrase,
    As 'Well, well, we know,' ...
    ...
    Or such ambiguous giving out, to note
    That you know aught of me: this not to do,
    So grace and mercy at your most need help you, Swear.

At first, the two men see no reason for swearing an oath but both Hamlet and the ghost insist. The ghost requires they swear:

GHOST[Beneath.]
    Swear.
[They swear.]

Hamlet then laments that he is alive during this time ("O time is out of joint"). It's the time in which his father was murdered and, being the king's son, he is the culturally required person to "set things right." But he is not happy about it and laments being born at the time--out of joint, out of proper alignment--that requires he avenge his father's murder.

Culturally, revenge killings were the norm, as illustrated by Fortinbras. It is Hamlet's relatively new religious beliefs--beliefs in time out of joint, time out of alignment--that make revenge killing something to lament: Fortinbras accepts revenge killing for his father/King's death as an honor.

Act 1, xcene 5 is the famous "Ghost Scene" in Hamlet, when the ghost of Hamlet's father, the former King of Denmark, appears to Hamlet to tell him the truth about his death (Hamlet's uncle, Claudius, did it!) and to implore Hamlet to swear that he'll avenge his death.

If thou didst ever thy dear father love—
Revenge his foul and most unnatural murder. (1.5.23-25.)

Caught up in the moment and eager to please his father, Hamlet swears to avenge his murder even before he's heard the story that his father came to tell him.

Haste me o know't, that I, with wings as swift
As meditation or the thoughts of love,
May sweep to my revenge. (1.5.29-31)

The Ghost tells Hamlet the story of his murder and makes his exit.

Adieu, adieu! Hamlet, remember me. (1.5.91)

Hamlet vows by heaven, earth, and hell to put everything aside until he has revenged his father's death.

Yea, from the table of my memory
I'll wipe away all trivial fond records...
And thy commandment all alone shall live
Within the book and volume of my brain,
Unmix'd with baser matter: yes, by heaven! (1.5.98-104)

Horatio and Marcellus come back to Hamlet to find out what happened in Hamlet's meeting with the Ghost. Hamlet makes a few cryptic remarks, then, with the help of the Ghost moving through the ground under their feet (1.5.148, 156, 161-162, 181), Hamlet makes Horatio and Marcellus swear on his sword, "Never make known what you have seen tonight" (1.5.144).

Hamlet also tells them that he's going to "Put an antic disposition on" (1.5.172)—in other words, he's going to start acting a little crazy—and that they must not pay any attention to his act or let anybody else know what's going on.

By now, Hamlet has calmed down from the excitement of seeing his father ghost and all the swearing, and he has a moment to think about the situation.

The time is out of joint: O cursed spite,
That ever I was born to set it right! (1.5.189-190.)

The meaning here is that time is like a bone that's slipped out of joint, and it's Hamlet's job "to set it right"—to set the bone back in place.

But Hamlet's clearly not happy about it. Denmark is a mess, and now he's got to clean it up. Here, "spite" might mean "revenge," and Hamlet is cursing the revenge that he has just sworn to undertake on his father's behalf. "Spite" could also mean that Hamlet feels that "the time," the world, and the universe are being spiteful towards him by forcing him into this situation. "Spite" might also mean his fate, and Hamlet is cursing the fact that he can't control it.

Just a few minutes after dramatically swearing by heaven, earth, and hell to avenge his father's death, Hamlet realizes the enormity of what he's tasked himself to do. Hamlet laments "that ever I was born," and he also resents the fact that his life is no longer his own, and it's now his sworn duty to revenge his father's murder and restore order to Denmark.

You have not included a question with this quote from "Hamlet".  I will make a leap of judgement and guess that you want to know a little about the quotation.  It is from Act 1, sc. 5 and it is at the end of the scene (and the end of the act).  Hamlet spoke the lines to Horatio and Marcellus.  Hamlet has just spoken to and heard from the ghost of his father who has asked Hamlet to swear to get revenge for his death.  The ghost told Hamlet that he was killed by his brother, the same brother, Claudius, who has married the former king's wife, Gertrude, and has now assumed the ghost's throne.  The lines set the tone for the rest of the play and Hamlet's reluctance and delay in getting that promised revenge.  Hamlet does not carry out the wishes of his dead father until the last scene of the play, and then, it is almost a spur of the moment, anger-driven act.  The lines indicate that Hamlet does not want this burden of responsibility ("cursed spite"), but he realizes that it is his duty as his father's son ("That I was ever born...") to get that revenge against his father's murderer.