In the epilogue of Deathly Hallows, when Harry’s son expresses trepidation about being sorted into Slytherin, Harry says:
“Albus Severus, you were named for two headmasters of Hogwarts. One of them was a Slytherin and he was probably the bravest man I ever knew.”
Bravery is the trait typically associated with Gryffindor.
If Snape was so brave, why was he sorted into Slytherin instead?
Snape wasn't sorted into Gryffindor for the same reason Peter Pettigrew was sorted into Gryffindor despite being a coward and Hermione was sorted into Gryffindor rather than Ravenclaw. Bravery, unlike intelligence, ambition, and kindness, is not a constant trait or characteristic. Everyone and anyone can be brave, thus anyone who asks to be in Gryffindor is placed in Gryffindor. Snape wasn't placed in Gryffindor because he was more suited to Slytherin at the time, and because he didn't ask to be.
Snape knew more curses when he arrived at school than half the kids in the seventh year and he was part of a gang of Slytherins who nearly all turned out to be Death Eaters.” – Sirius Black
The real question is, why didn't Snape ask to be in Gryffindor with Lily? Since E comes before S in the alphabet, Lily would have been sorted before Snape. If one can ask to be sorted into Gryffindor, and Snape loved Lily so much his Patronus matched hers, why didn't he ask the sorting hat to place him in Gryffindor?
The Sorting hat sorts you through a mix of your potential, your dominant trait at the time and your own preference.
There are plenty of examples of characters who lean one way or the other in this. Hermione would easily do as a Ravenclaw, but given that she has both intelligence and bravery, the fact that she values bravery more probably tipped the scales for her.
"Me!" said Hermione. "Books! And cleverness! There are more important things -- friendship and bravery and -- oh Harry -- be careful!"
Neville, on the other hand, shows little actual bravery at first - but we can infer he wishes to be brave, like his Grandmother. Of course, as early as the end of book 1 Neville has begun to show bravery, never mind by book 7 when he's a fucking badass shouting defiance to Voldemort's face.
"It takes a great deal of bravery to stand up to our enemies, but just as much to stand up to our friends. I therefore award ten points to Mr. Neville Longbottom."
The Hat seems to have valued his potential here, and possibly his (familial) preference.
And so we must consider young Snape. Let's go through his potential, dominant trait and preference.
Potential - Snape is cunning, resourceful and intelligent. Very gifted in practical magic. He also demonstrates incredible bravery later on in life. His potential leaves him open to Ravenclaw, Griffindor and Slytherin.
Dominant trait - Young Snape is dominated by his ambition and thirst for power, implied to be a direct result of his miserable home life with an abusive (Muggle) father.
...suddenly Harry’s mind was teeming with memories that were not his: a hook-nosed man was shouting at a cowering woman, while a small dark-haired boy cried in a corner... a greasy-haired teenager sat alone in a dark bedroom, pointing his wand at the ceiling, shooting down flies... a girl was laughing as a scrawny boy tried to mount a bucking broomstick -
"Oh yes, they’re arguing," said Snape. He picked up a fistful of leaves and began tearing them apart, apparently unaware of what he was doing. "But it won’t be that long and I’ll be gone."
It is true that Snape also shows some bravery in dealing with the Marauders, but this is a very blunt sort of bravery - note that he does not have the same fortitude when dealing with Lily and his Death Eater friends. He attempts to straddle both worlds for a while, unable to either face the evil of the Death Eaters, nor accept their creed entirely. Some moral development is still needed here. Based on this, his dominant trait clearly points him towards Slytherin.
Preference - Snape himself makes his preference perfectly clear on the Hogwarts express, when talking to Lily.
"You’d better be in Slytherin," said Snape, encouraged that she had brightened a little.
And when Griffindor is brought up by that prick James
Snape made a small, disparaging noise.
James turned on him. "Got a problem with that?"
"No," said Snape, though his slight sneer said otherwise. "If you’d rather be brawny than brainy-"
Beyond Snapes own statements, his actions up to this points indicate he's invested in Slytherin and it's affinity with the Dark Arts.
"Snape knew more curses when he arrived at school than half the kids in the seventh year and he was part of a gang of Slytherins who nearly all turned out to be Death Eaters."
Thus, his personal preference is clearly for Slytherin.
The basic conclusion here is that young Snape had the potential, the traits of, and valued Slytherin far more than Griffindor. The Sorting Hat made the correct choice for the young man. Of course, as we see in the books, Snape would grow into a far braver man than he'd have thought possible. All in all this can be summed up by the conversation between Snape and Dumbledore during the Yule Ball.
"Karkaroff’s Mark is becoming darker too. He is panicking, he fears retribution; you know how much help he gave the Ministry after the Dark Lord fell."
Snape looked sideways at Dumbledore’s crooked-nosed profile. "Karkaroff intends to flee if the Mark burns."
"Does he?" said Dumbledore softly, as Fleur Delacour and Roger Davies came giggling in from the grounds. "And are you tempted to join him?"
"No," said Snape, his black eyes on Fleur’s and Roger’s retreating figures. "I am not such a coward."
"No," agreed Dumbledore. "You are a braver man by far than Igor Karkaroff. You know, I sometimes think we Sort too soon..."
First, Severus Snape was eleven years old when he was Sorted to Slytherin; characteristics of each house are not exclusive. As J.K. Rowling writes, there are occasions when a Slytherin shows altruistic or selfless behavior (Snape is one example; another is Regulus Black, who sacrificed his own life in order to contribute to the cause of destroying Voldemort.). Accordingly, a Slytherin can be brave.
Technically, Harry says Snape was "probably" the bravest man he ever knew; this leaves a little wiggle room, that perhaps there could still yet be an individual out there, that Harry might encounter following his conversation with Albus Severus. It's a possibility.
Anyhow, regarding Snape's Sorting, at the time Snape was Sorted, the events that Harry considered as Snape's bravest acts had not yet occurred. According to J.K. Rowling, the Sorting Hat is skilled at Legilmency, but Legilimency does not allow the Legilimens to see into the future; it shows the Legiliments what is at time of the Sorting.
The Sorting Hat is notorious for refusing to admit it has made a mistake in its sorting of a student. On those occasions when Slytherins behave altruistically or selflessly, when Ravenclaws flunk all their exams, when Hufflepuffs prove lazy yet academically gifted and when Gryffindors exhibit cowardice, the Hat steadfastly backs its original decision. On balance, however, the Hat has made remarkably few errors of judgement over the many centuries it has been at work.
J.K. Rowling - Pottermore - Writings by J. K. Rowling
Note also that the Sorting Hat has made "remarkably few" errors in Sorting over the many centuries it's been at work.
Snape exhibited the traits we have come to know as associated with Slytherin House. There's no indication that Snape belonged in Gryffindor at the time he was Sorted.
I'm a bit late to answer here, but people can be suitable for more than one House. In those cases, the Hat seems to usually take people's preferences into account.
For example, Harry Potter was offered the choice to go into Slytherin but asked the Sorting Hat for "anything but Slytherin," which the Hat respected. Harry later tells his son that the Hat will take his preferences into account.
As others have indicated, Severus had indicated a strong preference for Slytherin before being sorted, indicating at one point that he hoped that Lily would be sorted into Slytherin like him. It's likely that the Hat respected that.
Also keep in mind that, in spite of being brave, he did have a significant capacity for bullying/cruelty as well, as evinced by the fact that he was apparently in Voldemort's inner circle at one point and his inclination to torment Neville and various other students. (Recall that Neville's Boggart was actually Professor Snape at one point).
It's also likely that that trait in Snape emerged over time. To quote the same passage as @DavidS,
"Karkaroff’s Mark is becoming darker too. He is panicking, he fears retribution; you know how much help he gave the Ministry after the Dark Lord fell."
Snape looked sideways at Dumbledore’s crooked-nosed profile. "Karkaroff intends to flee if the Mark burns."
"Does he?" said Dumbledore softly, as Fleur Delacour and Roger Davies came giggling in from the grounds. "And are you tempted to join him?"
"No," said Snape, his black eyes on Fleur’s and Roger’s retreating figures. "I am not such a coward."
"No," agreed Dumbledore. "You are a braver man by far than Igor Karkaroff. You know, I sometimes think we Sort too soon..."
This seems to imply that his bravery emerged after he was sorted into Slytherin.
It's also possible that the Hat will take family tradition into account. For example, when the Hat was placed on Ron Weasley's head, the hat remarked (and I'm heavily paraphrasing b/c I don't have the book in front of me) "Another Weasley! I know what to do with you - Gryffindor!" Of course, Ron was suited to the House anyway, so that's not conclusive, but it at least suggests the possibility.
How about Snape also having a preference for slytherin because his parents or at least his mother expects him to. She probably had been a Slytherin herself. He wouldn't want to cause more trouble or disappointment at home.
Another important factor he wanted to be in Slytherin was his mother. We can assume he had close relationship with her as he took her name as his nickname (Prince) showing he wanted to associate with her rather than with his abusive muggle father. We know his childhood passed in constant fights between his parents and he was glad to escape from this atmosphere. But the fact that he knew loads of information about Hogwarts leads us to conclusion that his mother spent a lot of time with him explaining about wizarding world. I think apart from Lily his mother was the only kind and important person in his life and he would ask the sorting hat to put him in Slytherin as it would make him emotionally closer to her.
Snape demonstrated many Slytherin qualities when he started at Hogwarts: an interest in the dark arts (knowing as many curses as a seventh-year, according to Sirius, plus his penchant for killing flies), a preoccupation and prejudice related to blood purity (disdain for Petunia due to her being a muggle) and a desire to join Slytherin (as expressed to James, Lily, and Sirius on the Hogwarts Express). At the time, he hated Gryffindor house and would not have wished to join it (which we learned in the same conversation on the train).
Snape continued to demonstrate all of these qualities during his school years at Hogwarts. His love for the dark arts led him to invent Sectumsempra and hang out with a bunch of future Death Eaters. He got so far into the whole pureblood mania that he called Lily a mudblood, and he continued to be loyal to Slytherin house. He also added the Slytherin ambition: mastering the art of potion-making (to the point that he could re-write the potions textbook) and giving himself the self-aggrandizing title of Half-Blood Prince, much as Tom Riddle did when he began to style himself Lord Voldemort toward the end of his time at Hogwarts. And, of course, he still hated Gryffindor.
Actually, even in the years leading up to his death, Snape was still very much the Slytherin. He covets the DADA professorship at Hogwarts, and works his way up to being school Headmaster and Voldemort's second-in-command. He favors Slytherins at every turn in his teaching. And the way he talks about the Dark Arts in book 6 is borderline pornographic. Oh, yeah, and he's constantly spiteful to Gryffindor students.
The only reason to even consider putting Snape is the courage he shows during the books, and off the top of my head I can't think of anything brave that Snape did until James & Lily were already dead, far too late to affect his sorting. Snape the schoolboy was about as much Gryffindor material as I am, and I'm the muggiest muggle that ever mugged!
Snape was put into Slytherin instead of Gryfindor because Slytherins tend to be very ambitious like how Snape wants to study Potions and teach defense against the dark arts and that's it. The other thing is that Slytherins have a very strong sense of self-preservation meaning that they don't just make reckless decisions, they think things thru and then act compared to Gryfindors who just act regardless of how dangerous the situation might be. Yes Snape is very brave but his self-preservation and ambition make him perfect for Slytherin.
