In the Harry Potter movies, the character of Severus Snape is played by Alan Rickman.
The enmity between Harry and Snape begins in Harry Potter and the Philosopher's Stone, from the moment that they first see each other. Snape has the physical appearance of a classic villain: tall, thin, hook-nosed and "oily," and clad from head to toe in forbidding black robes, "like a great black bat". Though Harry is repeatedly reassured that Snape is one of the professors guarding the stone, his estimations based upon Snape's appearance and his view of events leads him to believe that Snape is attempting to steal the stone.
However, further reading of this series of books reveals that appearances can be deceiving as we learn that Snape is actually one of the heroes. When Lord Voldemort rose to power the first time, Snape served him as a Death Eater; his left arm still bears the Dark Mark that showed his allegiance to his evil master. At some early point, however, it seems that he changed sides, and became a 'secret agent,' now answering Lord Voldemort's summons' to him through the Dark Mark for the purpose of spying on the doings of Voldemort and the Death Eaters and reporting their clandestine activities to Albus Dumbledore.
Much of Severus Snape's dislike of Harry Potter seems to arise from a rivalry between Snape and Harry's father, James Potter, when they were both attendees at Hogwarts. At first, it is hinted that Severus was obsessed with the Dark Arts, which seemed to put off James' clique. It was said that as a first year Hogwart's student, Snape knew more hexes and curses than most seventh-year students; furthermore, his bookish and "nerdy" demeanor and appearance made him an outsider and an object of scorn by the "popular" set.
In Harry Potter and the Prisoner of Azkaban, Harry is told that this dislike may be due to the perpetration of a dangerous "prank" by Sirius Black (James Potter's best friend), in which Black uses deceptive means to lure Snape to the Shrieking Shack where he would meet his death at the hands of Remus Lupin in his werewolf form. Snape's life is saved by James Potter; however Snape believes that Potter was actually involved in the plan to murder him and had only decided to rescue him out of fear of being expelled from Hogwarts. Later, in Harry Potter and the Order of the Phoenix, we further learn that Snape's dislike of Harry may also be the consequence of a grudge spawned during a humiliating public incident of unprovoked malicious maltreatment (hazing and bullying) suffered at the hands of James Potter and Sirius Black during their fifth year at the school (Snape's Worst Memory).
Despite Snape's intense dislike of Harry, he has saved the boy's life on more than one occasion; he is a member in good standing of the Order of the Phoenix, as is revealed in Harry Potter and the Order of the Phoenix, and as such has misdirected Dolores Umbridge in her attempts to discredit Harry. Despite this, he invariably gives Harry poor marks in Potions and harangues him verbally, and even attempted to have Harry (and Ron) expelled in Harry Potter and the Chamber of Secrets, after the flying car they had taken was seen by Muggles.
For a time during the course of the fifth book, Dumbledore assigns Snape the
task of giving Harry lessons in a branch of magic called Occlumency, having to deal with the protection of one's mind from outside intrusion. Snape is assigned this task because he is extremely skilled in both Occlumency and its companion art of Legilimency, both skills being undoubtedly useful for his undercover work among the Death Eaters. The classes were cut short, however, when Harry was caught using Dumbledore's Pensieve to observe Snape's Worst Memory. Viewing this memory proved to be an unpleasant shock for Harry, who discovered that his father had been an arrogant bully who persecuted Snape much as Dudley Dursley and Draco Malfoy persecute him. It seems that Harry's best strategy in dealing with Snape would be to emphasize their common ground and shared trait of being considered "outsiders." Only time (and Joanne Rowling) will tell if the two ever reach any kind of reconciliation.