<span style="background-color: black;color: black;">Harry is a Half-blood, Voldemort/Riddle is a half-blood, Longbottom is NOT halfblood (iir)</span>
Nature selects who will be a wizard at random. Hence you can have wizards born from families that never had a wizard in them before. There is no wizard race. Voldemort, Malfoys and the Death Eaters don't believe that. According to thier racist regime, if you were born from two muggles then you are really a muggle. Hence Mud Blood. If you were born from a muggle and a wizard, or a mudblood and a wizard you are half. If you are born from two wizards you are a fullblood. But this is all a fiction by the racist wizards. The truth is that wizardry is doled out randomly. The message I believe is that what you believe is your own and cannot be dictated by genes, or other people.
My guess is the half-blood prince is Dumbledore.
Aaron.