In the case of Snape. he was Sorted into Slytherin because he was ambitious, leaning towards the Dark Arts, his mother was one, AND he had an anti-Muggle bias, same as Salazar Slytherin himself.
TL;DR: Slytherin is a house for pure bloods, Tom Riddle and Snape were half bloods.
Students of any blood status could be placed in the house. However, a Muggle-born student from that house was considered to be quite rare.
Snape had a strong personal preference for Slytherin when he entered Hogwarts, which the Sorting Hat would have taken into consideration. Snape's personality conforms to the Slytherin values of ambition and cunning. Consider his potions textbook as evidence of how the schoolboy Snape thinks.
Voldemort recruited him and likes him because Snape had the same kind of feelings toward his Muggle father that Voldemort had toward his own. Snape's father was not a nice man. He was often abusive towards Snape's mother which is why he took the pseudonym “Half Blood Prince” to begin with.
In the second book, Rowling establishes that Voldemort hates non-pure-blood wizards, despite being a half-blood himself.
He would always resent Harry for being James' son instead of his. Snape even said that he didn't want anyone to find out that he was protecting Lily's child, “especially Potter's son” (pg. 679 of Harry Potter and the Deathly Hallows) because of his hatred for James.
But when they went to Hogwarts, everything changed. The pair were parted when the Sorting Hat placed Severus in Slytherin and Lily in Gryffindor, and while Lily thrived in her new, magical life, Snape was essentially placed in close quarters with the likes of Lucius Malfoy, Avery and Mulciber.
Snape uses his doe Patronus to show Dumbledore that he never fell out of love with Lily, his childhood best friend. Warner Bros.
But Harry did share some Slytherin traits: he was ambitious and cunning, always sneaking around after hours and cooking up plans with Ron and Hermione. His green eyes even match the house. It is curious to think what the Sorting Hat would have deduced without Voldemort's involvement in Harry's life.
He's a pure-blood wizard, proud Slytherin and Harry Potter's arch rival at Hogwarts. Happy birthday, Draco Malfoy!
The Weasleys
' We finish on a family who are technically quite pure-blood, but are deemed 'blood traitors' by various peers, and seem proud of it. It is, indeed, the Weasleys' compassion and morals that led to several members marrying half-bloods, Muggle-borns, etc.
Just like his father, Draco loved to hate on Muggles. 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.
Slytherin disliked taking students from Muggle families, seeing them as untrustworthy and unworthy of being taught magic, and tried to persuade the other founders to only take students from pure-blood families. However, the other founders didn't agree with Slytherin, particularly Godric Gryffindor.
He would have been more than welcome in Ravenclaw house. He was sorted into Slytherin because his mother was sorted there, and we all know the sorting hat takes your opinion into account. As a half blood, he was only exposed to the death eaters, and blood status seniority because he was in Slytherin house.
1 Albatross
As the rarest Patronus in the Wizarding World possible, the Albatross represents an enormous bird that flies freely across any sea. This rare Patronus represents those who are simultaneously fearless, optimistic, happy-go-lucky, and ambitious.
So when Ron said three, it wasn't a mistake. He meant there were three more horcruxes to destroy before being able to destroy the fourth and final piece which was in fact Voldemort himself.
The information in this broadcast will prove extremely heartening to Harry, as he learns that he is not alone in fighting Voldemort. It will, however, inadvertently result in the Trio being captured. We learn later that the doe was Severus Snape's Patronus.
At Hogwarts, she, along with her sisters, was sorted into Slytherin. It is suggested in the novels that, as a student, Bellatrix associated with a group of students – including Rodolphus Lestrange, Severus Snape, Avery, Evan Rosier and Wilkes – who nearly all became Death Eaters.
Voldemort intentionally made six Horcruxes, but when he used Avada Kedavra on Harry, he unintentionally created a seventh Horcrux. Instead of dying, Lily's love for Harry created a counter 'curse' known as Sacrificial Protection and saved Harry.
The girl was played by Chloe Rich in the film adaptation of Harry Potter and the Order of the Phoenix, and in Harry Potter and the Deathly Hallows: Part 2.
Here's the thing: Dumbledore knew that Snape was mean to the students, but the headmaster still allowed it. It's likely that he did this because he believed that his students needed life lessons, including how to deal with mean teachers.
He didn't tell him, because there was always a possibility of Voldemort peeking in Harry's mind and seeing that piece of information, because if voldy had learned that snape was a traitor, he would have killed him, and hid all the horcruxes into more safe places, that was the same reason why snape didn't gave harry the ...
To keep his cover pretty much. Bellatrix never trusted him and this was her way of trying to prove she was right. There is also the deal with Dumbledore that he must be the one to kill him, so It was pretty much already in action to happen anyway.