Because he wants to kill harry so he thinks he needs to have the "unbeatable" wand.
Voldemort believed the Elder Wand would make him invincible.
He wants to kill him so he could be the "master of the elder wand" (except for the fact that Snape was not ever the master of the elder wand....)
Lord Voldemort believes that Snape is the true master of the Elder Wand because Snape killed Dumbledore. In order to gain the wand's allegiance, he kills Snape. However, Draco Malfoy disarmed Dumbledore before that, so Snape had never been the true master.
Because Harry is the true master of the Elder Wand. Or because Voldemort's course didn't actually hit him, it fall back on Voldemort. I think this was because Voldemort couldn't kill Harry using a wand that he hadn't rightfully won? Or the wand didn't want to kill it's true master.
Draco was the actual master of the elder wand because Dumbledore's death was already planned, so when Snape killed him, it wasn't actually a takeover of the wand. Because Draco was the last to disarm Dumbledore, or 'take' his wand, it recognized Draco as it's master. This is how Voldemort was ultimately killed, because Harry was using Draco's original wand. The elder wand recognized the wand as its master's, so Voldemort's killing curse was turned back upon him. However, Draco never actually had the Elder Wand in his possession. He never even knew he was the master until Harry revealed it.
He was reduced to a powerless thing. The spell rebounded because Lily sacrificed her life, died for Harry, although she had the choice to live. Love is the strongest magic there is, and Lord Voldemort knew nothing of it because he had never been loved.
Albus Dumbledore defeated Gellert Grindlewald and became the owner of the Elder Wand. Draco Malfoy disarmed Albus Dumbledore in Harry Potter and the Half-Blood Prince which meant that Draco was recognised as the new owner of the wand. The only person who knew this was Dumbledore, who then died. At Malfoy Manor, Harry ran past Draco and pulled several wands out of his hand (making him the master of all of them) one of which was Draco's. Thus Harry was now the master of the Elder Wand without knowing it. Voldemort stole the Elder Wand from Dumbledore's tomb and thought that because he had taken it from him, he would be the master. But when he realized that it didn't work as well as it could for him he ordered his snake to kill Severus Snape who he thought was the true master. When Voldemort challenged Harry with the Elder Wand, it knew it could not kill it's own master so Harry's disarming spell worked and Harry caught the Elder Wand. Harry then decided that he didn't want the Elder Wand because it caused to much trouble, instead he repaired his old wand.
Harry and Voldemort find themselves facing each other for one final duel at the end of Deathly Hallows. Two curses only are cast. Harry casts Expelliarmus, trying to disarm Voldemort at the same time Voldemort performs Avada Kedavra (the killing curse). The jets of light collide and ricochet in reverse directions. Voldemort is hit by his own killing curse and dies. The answer is not this simple, though. Harry has to hunt down Horcruxes, visit places all over England, and read Voldemort's mind as part of his quest. Obviously you don't want to read the book yourself.
Hermione accidentally broke it when they were in Bathilda Bagshot's home, and Nagini attacked them. While she was apparating them, the wand snapped. Actually it got hit by one of Hermione's rebounding spells!
Snape doesn't want or need to get the Philosophers Stone, he is one of the staff protecting it Professor Quirrell wants the stone for his master (Lord Voldemort) He is sharing his body with Lord Voldemort.
Harry decides he doesn't want to keep it so he fixes his own wand and he returned it to Dumbledore's tomb.
Lord Voldemort did want to kill Harry. He promised that he wouldn't because Harry had the philosopher's stone and Voldemort wasn't powerful enough to take it off him, but Voldemort was unlikely to actually keep that promise even if Harry had joined his side and handed the stone over.The only reason Voldemort didn't try to kill Harry is because he had no body, he was sharing Quirrell's but had no actual control over him.