Pure-blood elitism. The most extreme of the pure-blood supremacists sought to purge the world of those they consider "impure". They often tolerated half-bloods out of necessity (though still disdained them), but Muggle-borns were persecuted and Muggles were viewed as nothing more than animals.
An interesting fact of note is that despite the ideals that pure-bloods are inherently more powerful wizards, several of the most powerful or particularly adept wizards and witches in the series are in fact half-blood, such as Albus Dumbledore, Tom Riddle, Severus Snape, Minerva McGonagall, and Harry Potter.
Though the family was inclined to blood-purity, they were unwilling to practise the dangers of inbreeding with cousins, correctly knowing that it would and could potentially endanger their genes, and would accept marrying half-bloods due to the small pool of available pure-bloods.
As such, he did not view half-blood witches and wizards differently from pure-bloods from a blood purity point of view, even if he considered them less respectable than pure-bloods, and thus Slytherin House let wizards such as Severus Snape and Tom Riddle into its ranks.
A Squib, also known as a wizard-born, was a non-magical person who was born to at least one magical parent. Squibs were, in essence, "wizard-born Muggles". They were rare and were looked upon with a degree of disdain by some witches and wizards, particularly pure-bloods.
Why the Death Eaters Accepted Voldemort as a Half-Blood - Harry Potter Theory
What happens if two squibs have a child?
Because, if you follow the definition to the letter, if a child has wizarding grandparents, then he IS considered a pure-blood, even if his parents are NOT wizards themselves. That would mean that someone born to two "pure-blood" squibs is a pureblood, even though both parents have no magical affinity.
Many older wizarding families placed great emphasis on blood purity and rejected association with Muggles and Muggle-borns, all of whom they considered greatly inferior to themselves.
He had no problem with half-bloods as long as they favored their wizarding heritage and didn't oppose him. Sure he considered pureblood to be higher than half-blood and would readily ignore his own muggle heritage, but he didn't have disdain for them in the way that had for muggle-borns and muggles.
Harry Potter had a few love interests during his time at Hogwarts. Though he eventually settled down with Ginny Weasley, he also had a thing for Cho Chang in his earlier school years. Then, of course, there's his secret love for Draco Malfoy… Wait, what?
Draco and his gang frequently antagonised Harry Potter and his friends, and Malfoy also used his position as prefect to bully some first years when he got the power in his fifth year.
Similarly, although disdaining half-bloods and considering them inferior, they recognised that there were not enough pure-blood families left and so (if only begrudgingly), accepted half-bloods in their new world order.
An interesting fact of note is that despite the ideals that pure-bloods are inherently more powerful wizards, some of the most powerful or particularly adept wizards and witches in the series are in fact either half-blood (such as Lord Voldemort, Albus Dumbledore, Minerva McGonagall, Severus Snape and Harry Potter) or ...
The pure-blood wizards and witches featured in the Harry Potter books are almost all supremacists, while there are some of them who do not advocate ancestral superiority; the Potters, Weasleys, and Longbottoms are old pure-blood families, but no known members of these families are sympathetic to supremacist aims.
How can Snape be in Slytherin if he is half-blood?
Because Slytherin was NOT the house for ONLY pure-bloods. If that were the case, then Slytherin would NEVER have a full house. Look, Slytherin house IS willing to accept half bloods. There are even reports of rare muggle borns entering their house.
With no one to claim him as their own, Tom lived in a Muggle-run orphanage where he felt abandoned and unloved. He slowly realized he was different from others and became obsessed with his powers, believing he was unique and elite.
Echoing the views of Salazar Slytherin, one of the founders of Hogwarts, Voldemort believes that non-magical ancestry somehow taints the bloodline. Only those with no such taint deserve to be called Pureblood, and only Purebloods deserve to be taught magic.
Warner Bros. Snape was a half-blood, born to a Muggle father named Tobias Snape and a witch mother named Eileen Prince. His father was neglectful and sometimes abusive, which may have contributed to Snape's disdain for Muggles.
He never knew that his mother was a wizard. He thought she must have been a muggle, otherwise she wouldn't die like that. He didn't like muggles because they didn't have magic. For him, that was equivalent to being weak.
Once at school, Tom committed himself to learning about his parents, only to arrive at the disappointing conclusion that his mother was an unremarkable witch and his father was a Muggle. This was when he abandoned his birth name in favor of his new name: Lord Voldemort.
Dolores Jane Umbridge was born to a Wizard father and a Muggle mother, and eventually had a younger brother who was a Squib. Under the influence of her father, Dolores grew up despising her mother and brother for their lack of magical abilities.
The Malfoy family, known for despising Muggles and Muggle-borns, actually gained property and wealth through interactions with Muggle royals. Draco Malfoy, influenced by his prejudiced father, eventually grows out of his family's pure-blood mania, which was ingrained in the Malfoys' ethos.
He took particular pleasure in taunting Hermione Granger, who just happened to have Muggle parents. Malfoy called her 'Mudblood', a very grave insult referring to a wizard or witch born to non-magic parents.
