Harry & Snape
I've always liked the inteaction bewteen Harry and Snape, how Harry is always trying to make him the bad guy and how Snape delights in using his power as a Professor over Harry.
How? What? Why?
NOTICE: This section contains information and speculations based on all SIX Harry Potter books. If you do not wish to be spoiled, espcially concerning Harry Potter and the Half Blood Prince, do not read.
Despite my liking for imagining Snape and Harry enjoying their oh so recurrent detentions together, most of the following is strictly canon, because, let's face it, nothing will ever happen between them in the books! The fun is, I'm afraid my friends, solely possible in fanfiction!
From the moment their eyes first connected, they hated each other. Harry's scar tingled (although it was most likely to have something to do with silly old Professor Quirell sitting next to Snape!) and Snape had a tweenty year old prejudice, no matter what Harry could possibly do.
During their first Potions lesson, Snape doesn't even try to hide his loathing for Harry and shows him up in front of the class. Snape's behaviour towards Harry has always been nothing but unfair. And Harry is always first to blame Snape whenever something shady is going on.
In year one, Harry is sure Snape is trying to get his hands on the Philosopher's Stone - and kill Harry - but it turns out Snape was in fact trying to stop Quirrell from taking it. And he actually saves Harry from Quirrell's attempts to kill him.
Several times arry tries to warn Dumbledore about his doubts on Snape's allegiances, even more so in the Half-Blood Prince, but no one will listen.
Harry's hatred for Snape reaches it's height when he finds out that Snape was the Death Eater that spied on Trelawney & Dumbledore when the prophecy was fortold and that he reported what he heard to Voldemort, leading to the attack on Harry and his parents.
Everybody believes he's on the right side, except Harry, until Snape kills Dumbledore, using the Avada Kedavra curse.
After Dumbledore's funeral, Harry sets out to destroy Voldemort, prepared to kill Snape on the way.
As for the whole 'Snape's betrayal' thing, I have to admit I'm confused. Part of me is inclined to say that he fooled Dumbledore (Dumbledore actually mentions that he makes mistakes and when he does, they're pretty huge ones too...), but it does indeed seem too simple, and there might be more to it than we think.
Severus Snape and... Lily Evans?
I don't really ship the two, but it's something that sprung to mind. Snape hates James to an extreme extent, and, yes, Snape despises non pure-blood wizards, but I don't know, it seems to make sense that Snape may have had an unspoken love for Lily.
As Slughorn pointed out in HBP, Lily was totally lovable, kind and pretty. I don't think anyone have ever spoken ill of her. Snape has never actually even mentioned her, and given his character, he's not the kind to suppress a snide comment. So, just a thought really...
Snape's Redemption
I doubt going up to Dumbledore and telling him "I'm sorry I screwed up. I think I was wrong about the whole 'let's be Voldemort's follower and kill things' part" would convince him to trust someone like Snape, especially to the extent Dumbledore does. He trust him with serious and important missions. He wouldn't do so if he wasn't sure of his loyalty.
Reading HBP, I wondered, did Dumbledore make Snape perform an Unbreakable Vow? Maybe he made sure he would always protect Hogwarts students above all things? His action throughout the books always show him protecting the students the best he can, even when he truly hates them. He protected Harry during his first Quidditch game, he protecte dthe trio from the werewolf, he protected Draco, preventing him from becoming a killer, and he protected Harry, because he refused to fight him as he fled with Draco (also protecting Draco again). All this is just speculation, but it does seem to make sense to me!
Snape & James
I think it's important here to talk about Snape's relationship with Harry's father here, since - at the beginning at least - it's the main reason of Snape's dislike for Harry.
When they were at school, Snape and James hated each other. It was common knowledge and they didn't hide it. They were everything the other hated. Snape was a Slytherin, obsessed by the Dark Arts and convinced that Muggle-borns were inferior. James was a popular, somewhat arrogant Quidditch player, favored by students and staff and who always seemed to get away with everything.
It does seem that when they were at school, James had the upper hand and regularly humiliated Snape. When he becomes the teacher and sees Harry sitting in front of him, I think he sees James all over again. Harry is everything he hated about him and more.