One of the underlying problems in the wizarding world is intolerance and prejudice based on the "purity" of a person's wizarding blood. Many members of the old, pure-blood families look down on those who are not pure-blood. This intolerance permeates the entire culture and also extends in many cases to a prejudice against "half-breeds," such as the half-giant Hagrid and creatures such as centaurs. Voldemort exploits this prejudice, turning the natural enmity between various groups in the wizarding world to his advantage. Many people, such as Sirius Black's parents and brother, were first drawn to support Voldemort because he preached a message of intolerance toward non-pure-bloods.
Interestingly, Voldemort, who pushes a pure-bloods philosophy, is a half-blood himself. It is possible that his followers do not realize this, since Bellatrix acted as though Harry were uttering blasphemy when he mentioned that fact to her. It is also interesting, according to Dumbledore, that when Voldemort chose which child to attack of the two who fit the prophecy, Voldemort chose the half-blood like himself as being the more dangerous (OP37).
In order to understand the entire issue of blood status, it is necessary to understand the nature of wizards.
There are terms used, some more polite than others, for people depending on their blood status.
pure-blood families are extremely rare, and thus are interrelated; given the limited selection of potential pure-blood spouses, pure-bloods are getting rarer as their members either become more liberal and marry out, so that their descendants are no longer pure-blood, or consciously restrict themselves to pure-blood marriages.
Note that there is nothing good or bad about being pure-blood, in and of itself. The only issue with maintaining pure-blood purity is that it requires either a lucky chance (falling in love with someone who happens to qualify as a fellow pure-blood) or deliberately maintaining a prejudiced attitude about who is acceptable. The latter characteristic is why so many of the pure-blood families we know of find Voldemort's philosophy attractive - many have deliberately chosen a course of action for generations that is compatible with his, to a certain degree.
Both Harry Potter and Tom Riddle (now Lord Voldemort) are half-bloods. Harry taunted the Death Eaters with this in the Department of Mysteries to stall for time (and as a side benefit, making the Death Eaters so angry that some of them lost the fine edge of their self control):
"Did you know he's a half-blood too?" said Harry recklessly. Hermione gave a little moan in his ear. "Voldemort? Yeah, his mother was a witch but his dad was a Muggle - or has he been telling you lot he's pure-blood?" (OP35)
After Voldemort and his minions assassinated Minister of Magic Scrimgeour during the Second Wizarding War and took over the Ministry of Magic, Muggle-borns were required to present themselves to the Muggle-born Registration Commission and were listed in the Muggle-born Register (DH11).
pure-blood who doesn't insist on maintaining his or her purity.
Essentially, the term is used (generally by prejudiced pure-bloods as an insult, but occasionally ironically by those to whom it applies) to refer to a pure-blood witch or wizard who doesn't subscribe to the bigoted extreme of pure-blood-consciousness. Sirius was considered a blood traitor by his parents, and as he said to Harry while looking at his mother's edited tapestry of the Black family tree,
"...if ever a family was a bunch of blood traitors it's the Weasleys."
A Squib is not a Muggle. Born to a wizarding family, a Squib has such a low level of magical power that he or she is essentially unable to do any magic at all. However, while a Squib cannot cast spells, he or she can apparently see magical beings such as poltergeists, though not dementors (JKR).
Some Squibs seem to have formed special bonds with cats, whom they refer
to as Mr or Mrs. It is possible that these cats function as guides and
aides to Squibs as they live in a world in which they don't really fit.
In a sense, these cats may be the wizarding equivalent of Guide Dogs
and other animals which are trained to help Muggles with disabilities.
J.K.Rowling on blood status, prejudice, and intolerance:
Entertainment Weekly, 9/7/2000
Because bigotry is probably the thing I detest most. All forms of intolerance, the whole idea of "that which is different from me is necessary evil." I really like to explore the idea that difference is equal and good. But there's another idea that I like to explore, too. Oppressed groups are not, generally speaking, people who stand firmly together -- no, sadly, they kind of subdivide among themselves and fight like hell. That's human nature, so that's what you see here. This world of wizards and witches, they're already ostracized, and then within themselves, they've formed a loathsome pecking order.
From the beginning of Philosopher's Stone, prejudice is a very strong theme. It is plausible that Harry enters the world wide-eyed: everything will be wonderful and it's the sort of place where injustices don't happen. Then he finds out that it does happen and it's a shock to him. He finds out that he is a half-blood: to a wizard like Lucius Malfoy, he will never be a true wizard, because his mother was of Muggle parentage. It's a very important theme.Voldemort's a half-blood too...
Like Hitler! See! I think it's the case that the biggest bully takes their own defects and they put them on someone else, and they try to destroy them. And that's what he [Voldemort] does. That was very conscious - I wanted to create a villain where you could understand the workings of his mind, not just have a 2-D baddie, dressed up in black, and I wanted to explore that and see where that came from. Harry in Book Four is starting to come to terms with what makes a person turn that way. Because they took wrong choices and he [Voldemort] took wrong choices from an early age. (Newsround)from www.jkrowling.com:
Why are some people in the wizarding world (e.g., Harry) called 'half-blood' even though both their parents were magical?
The expressions 'pure-blood', 'half-blood' and 'Muggle-born' have been coined by people to whom these distinctions matter, and express their originators' prejudices. 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 mother's grandparents.