"So the oldest brother, who was a combative man, asked for a wand more powerful than any in existence: a wand that must always win duels for its owner, a wand worthy of a wizard who had conquered Death! So Death crossed to an elder tree on the banks of the river, fashioned a wand from a branch that hung there, and gave it to the oldest brother."
-- The Tale of the Three Brothers (DH21)
Also known as the Elder Wand or Wand of Destiny, the Deathstick is one of the three Deathly Hallows. Made of elder with a Thestral tail hair core, the wand is unbeatable in the hands of its rightful owner.
The Elder Wand, originally owned by Antioch Peverell, has the most storied history of the Hallows. It has cropped up through the centuries under different names, including the Deathstick and the Wand of Destiny (DH21). It was given the name Deathstick by the Dark wizard Loxias, in the early eighteenth century (TBB). At one point it was owned by Gregorovitch, the wand-maker who made Viktor Krum's wand, and stolen from him by a young, blond Grindelwald (DH18), though at the time Gregorovitch did not know who he was (DH14). This most likely happened sometime in the early twentiety century (see the timeline for more).
As Grindelwald gained power, word got back to Dumbledore in Britain that he had obtained a "wand of immense power" - the Elder Wand (DH35). He used it for destructive purposes, though, and in 1945 Dumbledore challenged him to a duel and defeated him, landing Grindelwald in prison for the rest of his life and winning Dumbledore both widespread notoriety and the Elder Wand (PS6, DH35). Dumbledore used the wand until his death in 1997.
Upon his death, Dumbledore intended for Severus Snape to become the master of the Elder Wand, but failed to take into account that, as his death was pre-planned, Snape had not truly defeated him. Instead the wand's rightful master became the wizard who had disarmedDumbledore just prior to his death: Draco Malfoy. When Harry stole Malfoy's wand during a confrontation in the Malfoy Manor (DH23), he became the master of the wand, though neither he nor Malfoy had ever touched or used it (DH36).
The wand itself was buried with Dumbledore and stolen from his grave by Voldemort (DH24). Voldemort, however, never became the true master of the wand. Believing that the master was Snape, he killed him (DH32) before his final confrontation with Harry, when he died and the wand passed into Harry's possession once and for all (DH36).
Harry only used the Elder Wand for one spell - Reparo - when he used it to fix his own broken wand. Beyond this he intends to die naturally without using it, instead returning it to Dumbledore's tomb in the hopes that there will never again be a true master of the Elder Wand, and that therefore its powers will be depleted (DH36).
Thestral tail hair (JKR, TLC2)