Voldemort killed Lily and James Potter because he feared that their son Harry would someday become a threat to his power. He got this idea because Severus Snape told him of the Prophecy about the Dark Lord and either Harry or Neville. The prophecy stated that the one who would destroy the Dark Lord would be a boy born in the month of July whose parents had defied the Dark Lord three times. This description fit both the Potters and the Longbottoms, and both Harry and Neville were born in July. What Voldemort missed was that he would mark the child as his equal and he would grow to have greater power. Voldemort ended up choosing Harry, who was a half-blood like him, rather than Neville Longbottom, a pure blooded wizard. Dumbledore informed Harry of the power that he had over Voldemort: the ability to love. Voldemort realized it was his weakness entirely too late.