Half Blood, Pure Blood--Help Pease

[sro35 at webtv.net]

No: HPFGUIDX 129783

I do not understand what makes a person half blood. In OOTP, DD 
tells Harry that Voldemort chose him, the half blood, and not 
Neville, the pure blood. What makes Harry a half blood?

Both of Harry's parents were magical, doesn't that make him pure 
blood? Does Lily coming from a Muggle family change it? If so, 
why?

Thanks in advance for your help....Stan (anxiously waiting for 
July 16)