Severus Snape

Severus Snape is best known as the Head of Slytherin House and Potions Professor at Hogwarts during Harry Potter's time at school. He also taught Defense Against the Dark Arts for one year.

Character
He is described as a batlike, greasy, hooked-nosed man -- sometimes hovering ominously like a vulture over certain unfortunate students. In the bookverse, he is coldly angry and mean to most of his students, and especially to Harry Potter. In the early books, he was a clear red-herring 'villain' that young Harry Potter and his friends were suspicious of. His character was later expanded to reveal that he has a long-standing hatred and resentment of his childhood, and anything that reminds him of it is an object of his ire.

Snape is a Death Eater and a double agent for both Voldemort and Dumbledore. He is a master of Occulumancy, which makes sense or else Voldemort would have noticed his double agent act and killed Snape right away.

In the movieverse, his cold resentment is portrayed sometimes as a smug dominance -- which may explain why so many fangirls fancy him even without being portrayed by Alan Rickman.

Relations
Snape's history very much defines his character. He is the head of Slytherin house for Harry Potter's first six years at Hogwarts and is very much biased in favor of his house.

As mentioned above, Snape resents Harry Potter for being the son of Lily Evans, who he had romantic feelings for and never married. However, he also protects Harry Potter at times for this same reason.

Snape hates all of the Marauders because they teased and tormented him all through school. He especially hates James Potter because he was both the most cruel to him, and he managed to marry the woman he loved.

Snape is mean to Neville Longbottom because Neville is bad at potions.

Snape dislikes Griffindors in general and even will abstain from giving Hermione Granger points for brilliance.

Snape is a double agent for Dumbledore because Voldemort killed Lily Evans.

In Badfic
Snape is popular. Really popular. Popular to the point of being one of the most likely characters for an Author-Wraith to possess. For some reason, despite his greasy appearance and mean demeanor (and possibly because of Alan Rickman) he is portrayed as a memetic sex god. He is paired with everybody. Everybody. Even when it would be really, really creepy to do so.

In Canon, he is mean and often pushed too far, but never evil. Some writers like to make him evil to bully another character, and show off that other character as a woobie.

For some reason, some individuals are under the impression that he would make a good father or mentor figure for Harry Potter. This makes no sense because he hates Harry Potter's guts and gets into such danger as being a Death Eater double agent and thus would be totally unsuitable.

Others make Snape wangst. They make him wangst on his past far harder than he ever did in the books or movies, and frankly this is really annoying and bad; Snape never really was a woobie, and he never should be.