The internet is dark and full of terrors. Some of those terrors come in the way of fandom that imagines Harry Potter as a member of House Slytherin. In response to those Tumblr screenshots I always see, I wanted to take a minute to discuss why Harry Potter wasn't sorted into Slytherin in the first place.
When Voldemort attacked the Potter family his killing curse was rebounded by, of all things, Lilly Potter's love for her child. While the rebounding spell had a lot of terrible consequences for the Dark Lord, the worst thing that happened was that a bit of Voldemort's already fragmented soul was transferred to Harry. This is why Harry has the ability to speak with snakes, why he has a lightning bolt shaped scar on his forehead, why he has the wand that he uses, why everyone knows his name, and why he must die at the end of the series.
When the Sorting Hat is placed on Harry's head it immediately begins searching for qualities that make him suitable for one of the four Hogwarts houses: Gryffindor, Hufflepuff, Ravenclaw, or Slytherin. The qualities that Sorting Hat finds, though, are the ones imparted on Harry by the presence of Voldemort's soul. That is why the Sorting Hat considers putting Harry in Slytherin at all- because it senses the great talent of Tom Riddle.
The qualities of the boy Harry Potter are so slight and insignificant that they were getting drowned out by even a shard of Lord Voldemort's soul. Harry Potter did not get sorted into Slytherin because Harry Potter is unworthy of Slytherin house- he is not cunning, ambitious, or resourceful. Harry was placed in Gryffindor because the Sorting Hat couldn't find any special qualities to designate him anywhere... so it simply granted his request to be with his friends.
Find below a YouTube clip of the Sorting Hat scene from Harry Potter & The Sorcerer's Stone, and a short transcript of the events.
"hmm.. difficult, very difficult. Plenty of courage I see. Not a bad mind either. There's talent, oh yes, and a thirst to prove yourself. But where to put you?"
This is where Harry begs the Sorting Hat by whispering the words "Not Slytherin".
"Not Slytherin, eh? Are you sure? You could be great you know, it's all here in your head, and Slytherin will help you on the way to greatness. There's no doubt about that. No? Well, if you're sure, better be... Gryffindor!"
It's also worth noting that the rest of the series, not just the Sorting Hat scene, depicts Harry as more of a liability than an asset. His friends are always rallying to save his life and defend him against the consequences of the rules he breaks. Ron and Hermione follow Harry into disastrous situations and rely on their own skills, knowledge and courage respectively, to find a way out.
At the beginning of the series the only thing unique about Harry is the level of attention he gets from the wizarding community and the way trouble constantly finds him- and both of those qualities are thanks to Voldemort. As the series goes on, Harry's experience builds him an army of students that want to learn his defense against the dark arts skills- but Harry even has Voldemort to thank for putting him the situations where he developed those skills. Without Voldemort, who is Harry Potter?