Harry's magical parents are Lily Evans (a muggleborn) and James Potter (a pureblood). Since a pureblood is someone with magical parents, why isn't he considered a pureblood?
A muggleborn, or mudblood is a person with powers from an ancestor. A halfblood is a person with one parent that is magical. Given that Harry has two magical parents, why is he only a "half-blood"?
It's a matter of definition, and hearkens back is similar to the 'One drop Rule' that used to be well known in the US. Basically, to be a pureblood, you must come from magical families on both sides, not just two magic-using parents. Conversely, any muggle blood makes you 'half-blood' at best.
From Pottermore:
"The term 'pure-blood' refers to a family or individual without Muggle (non-magic) blood. The concept is generally associated with Salazar Slytherin, one of the four founders of Hogwarts School of Witchcraft and Wizardry, whose aversion to teaching anybody of Muggle parentage eventually led to a breach with his three fellow founders, and his resignation from the school."
None of the families are really pure-blooded, as Dumbledore points out in Beetle the Bard
"So-called pure-blood families maintain their alleged purity by disowning, banishing, or lying about Muggles or Muggle-borns on their family trees. Then they attempt to foist their hypocrisy upon the rest of us by asking us to ban works dealing with the truths they deny. There is not a witch or wizard in existence whose blood has not mingled with that of Muggles..."
Harry's Mother's family is not a recognized wizard family, so he has muggle-blood. Any muggle-blood with wizard blood defines him as a Half-blood.
To go straight to the horse's mouth, here's JK Rowling on the subject:
(Note: This is from the Harry Potter Wiki, and I haven't been able to substantiate it as an actual JKR quote yet.)
The expressions ‘pure-blood’, ‘half-blood’, and ‘Muggle-born’ have been coined by people to whom these distinctions matter, and express their originators’ prejudice. As far as somebody like Lucius Malfoy is concerned, for instance, a Muggle-born is as ‘bad’ as a Muggle. Therefore Harry would be considered only ‘half’ wizard, because of his maternal grandparents. If you think this is far-fetched, look at some of the real charts the Nazis used to show what constituted ‘Aryan’ or ‘Jewish’ blood...the Nazis used precisely the same warped logic as the Death Eaters. A single Jewish grandparent 'polluted' the blood, according to their propaganda.
On account of his mother, Lily.
A pureblood is someone with magical parents...A halfblood is a person with one parent that is magical.
You're kind of right. The criteria for Muggle-born and pureblood are pretty straightforward. Rowling is much more vague and inconsistent when it comes to half-bloods. You can be a half-blood either if one of your parents was a Muggle (and the other a witch/wizard) or if one of your parents was a Muggle-born witch/wizard. What matters is that at least one parent has some connection to the Muggle world, whether they remained a Muggle until their death or they later went on to become a witch or wizard.
In Harry's case, he counts as a half-blood on account of his mother Lily. She was a Muggle-born witch with two Muggle parents. That was enough to make Harry a half-blood.
