Well, this wand belonged to Grindelwald. Dumbledore defeated Grindelwald in their epic battle, so he got the wand. Voldemort aliiba the wand from Dumbledore's grave, thinking he was the owner since he had defeated the aged wizard. However, the wand wasn't quite working for him. So he realized that it was Snape who had technically killed Dumbledore, so he must have been the true owner. That's why he killed Snape. What Voldemort didn't know was that Malfoy had disarmed Dumbledore long before Snape arrived in the tower and killed the headmaster. Therefore, Malfoy was the true owner. In the Malfoy Manor, however, Harry defeated Draco in a battle. So he eventually became the final owner of the elder wand.