Wizarding bloodlines work sort of like this:
Let's say your mom is half mexican/half irish, and your dad is full mexican. You wouldn't consider yourself completely mexican, would you? No, you would be considered a half-blooded mexican or half-blooded irish. (technically, it would be three quarters, but that doesn't sound as nifty).
Here's how I see it:
Lily and Petunia were born to two Muggle parents, right? Which in my mind says that somewhere along the line, a Squibb (person born to a witch and wizard that has no magical ability, or cannot use their magic for some reason) was born out of a magical bloodline. That Squibb moved on to marry other Muggles or Squibbs, which resulted in the magical gene becoming dormant (unused).
Then, when Lily was born, POOF, it came back to life and she happened to be the one to get it. Which means she's a Muggleborn, or someone with less than half magical blood. Kind of like Hermione.
James Potter is a pure-blood, because both of his parents were at least a half-blood or more. (AKA, his mum had two magical parents and so did his dad).
That means that when they had Harry, technically he would be a three-quarter-blood, but you get the idea.
The reason? You can either do magic or you can't. Unless you're a Squibb (who might as well be Muggles who sometimes get it right), there's no gray area. You can either do spells and make potions and go to Hogwarts, or you're a Muggle.
The less insane version.
Mugglborn-someone with two magical parents and no known magical ancestors.
Half-Blood- someone with at least one non-magical parent or grandparent.
Pure-blood- someone with no known muggle ancestors at all.