Half-breed

"Well, I think this should put an end to the oaf's teaching career. Half-giant...and there was me thinking he'd just swallowed a bottle of Skele-Gro when he was young... none of the mummies and daddies are going to like this at all... they'll be worried he'll eat their kids, ha, ha..."

- Draco Malfoy after Hagrid's half-breed status is exposed

Half-breed is a term given to humans with at least one non-human parent, although those with more distant non-human ancestry are also referred to as half-breed.

"Half-breed" may be an offensive, rather than proper, term, as it seems to appear as an insult. There is a great deal of prejudice against half-breeds in the wizarding world, which also tends to discriminate against non-human beings in general.

Traits
Half-breeds share the attributes of both their parents, still being able to use magic, as the magic gene is dominant, while having traits of their non-human parent as well. It is also unknown for how many generations non-human traits will persist, although it seems that wizards and witches with even one non-human great-grandparent will retain some of those characteristics. For example, Filius Flitwick is only one-eighth Goblin, but retains a small stature as a result of this heritage.

It is unknown if Muggles can interbreed with non-humans and, if they can, if this would result in non-magical half-breeds. Though it is very rare for a Muggle to even find out about the existence of those creatures, so the existence of a non-magical half-breed is unlikely.

Attitude Towards Half-Breeds
Many wizards and witches have a prejudice against non-human beings, from house-elves to goblins to Centaurs, and extend this prejudice to those with mixed heritage. An example of this prejudice would be Dolores Umbridge, and her decrees to continue to suppress beings whom she deems are not as high enough esteem.

When Rubeus Hagrid's status as a half-giant was exposed by Rita Skeeter in 1994, many parents of his students at Hogwarts were alarmed, assuming that Hagrid must be savage and dim-witted because of his ancestry.

It is also possible that there is less prejudice toward different types of half-breeds, as part-Veela Fleur Delacour never seemed to suffer any discrimination based on her heritage. This may be because there is no clear physical difference between humans and Veelas, and because even part-Veela possess stunning good looks and the ability to entrance the opposite sex supernaturally, where as a Hagrid and his Giant appearances is less appealing than Fleur and her Veela appearances.

Half-Giant
A half-giant is the offspring of a giant and a human. Half-giants will be much taller and larger than ordinary humans, will have considerable spell resistant skin, and they will possess great physical strength, though not to the same extent as giants.

Rubeus Hagrid is a half-giant, with a wizard father and giantess mother. The exact parentage of Olympe Maxime is unclear; weather she had a Giant father or mother is unknown.

Half-Goblin
A half-goblin is the child of a goblin and a human. Half-goblins are much shorter than ordinary humans, and are likely more clever than most humans as well, a trait they could inherit from their goblin ancestry.

There are no half-goblin characters in the series, but Filius Flitwick has "a dash of goblin ancestry", meaning that one of his ancestors was a goblin.

Half-Veela
A half-Veela is the child of a wizard and a Veela, which as far as we know is typically a girl. Because Veela are always female, a half-Veela will always have a human father and a Veela mother. Half-Veela are considerably more beautiful than average humans, and seem to possess the Veela power of entrancing men. It is unknown if part-Veelas are able to transform into bird-like creatures, as full Veelas can.

It was, at one time, unknown if a being with Veela blood could produce a male, as all examples of part-Veelas were female. However, quarter-Veela Fleur Delacour and wizard Bill Weasley had a son named Louis. It is unknown if he is able to entrance human females the way female part-Veelas entrance male humans.

Apolline Delacour is a half-Veela, making her daughters Fleur and Gabrielle quarter-veelas, and her grandchildren, Victoire, Dominique, and Louis Weasley one-eighth Veela.

Other Half-Breeds
It is possible that there are other types of half-breeds. Harry Potter once thought to himself that Marcus Flint looked like he might have had troll blood in him, though he may have simply been insulting him, as Marcus was large and rather ugly. Ron Weasley also once stated that a Snatcher was “definitely part troll” based on his body odour, though he was intentionally being humourous and thus might not have meant this literally.

Other human-like "creatures" include Merpeople, vampires and Centaurs, though it is unknown if these people can, or will, interbreed with humans.

The singer Lorcan d'Eath was described as being "part-vampire". This presumably indicates that he had at least one ancestor who was a vampire.

Mistaken for Half-Breeds
Centaurs are their own non-human breed, even though they look to be half-human. They are not half-humans, half-horses, even though they would be described that way. They find it extremely offensive to be referred to as half-breeds, as seen in 1995 when Dolores Umbridge insulted a heard of centaurs by calling them a half-breed.

Werewolves are not half-breeds; they are human victims of the magical affliction lycanthropy. It is unknown if "half-werewolves" exist; the only close example is Teddy Lupin, the son of a werewolf and a witch. He is a Metamorphmagus like his mother. Considering his father's worry that he would pass on lycanthropy to his child, it seems possible that the affliction is somewhat hereditary.

One human parent, one non-human parent

 * Rubeus Hagrid — pure-blood wizard father, giant mother
 * Olympe Maxime — wizard or witch parent, giant parent
 * Apolline Delacour — wizard father, Veela mother

Two magical parents, known non-human ancestry

 * Lorcan d'Eath — one vampire ancestor (assumed)
 * Fleur Delacour — one Veela grandmother
 * Gabrielle Delacour — one Veela grandmother
 * Filius Flitwick — one goblin ancestor
 * Dominique Weasley — one Veela great-grandmother
 * Louis Weasley — one Veela great-grandmother
 * Victoire Weasley — one Veela great-grandmother