The Wand

It actually is quite confusing... yes Dumbledore owned the Elder Wand (movies don't explain how) and it was buried with him. Snape killed Dumbledore and presumably took ownership of the Elder Wand, however since Draco Malfoy actually disarmed the wand from Dumbledore, the wand itself considered Draco its true owner even though Draco never held it. When Harry Potter disarmed a different wand from Draco, the Elder Wand somehow considered Harry Potter now the true owner, even though the wand was either still with Dumbledore's body or already stolen by Voldemort. Once he had the Elder Wand, Voldemort killed Snape thinking the wand would align with him, not knowing it was never truly Snape's (Snape knew all this and played along to protect Harry, one more heroic act for him). In the final battle with Harry Potter, the wand refused to kill its true owner, Harry Potter, and failed Voldemort leading to his defeat.

So Harry accidentally won by taking a random wand from Draco...

Throughout the movies (the two Deathly Hallows movies), Voldemoret is looking for the Elder Wand. He begins with the wand maker, Gregorovitch, who says someone stole it. That someone turns out to be Grindelwald. In the first movie, we learn from his chocolate frog card that one of Dumbledore's major accomplishments was defeating Grindelwald. It was then that the Elder Wand became his.

So, yes, the Elder Wand had been Dumbledore's for a long time.