In the fifth Harry Potter book - The Order of the Phoenix - Snape arrived at Grimmauld Place on Dumbledore's orders to see Harry and to let him know he would be receiving private Occlumency lessons with him once a week. Harry, who'd never before heard the word 'Occlumency,' was immediately informed it was the magical defense of the mind against external invasion. While Harry protested, Snape maintained the lessons were being arranged because the headmaster thought they were a good idea.
Snape also dismissed Harry's godfather, Sirius Black, who was present at this meeting, with terse remarks, signifying that their rivalry lived on. Sirius warned Snape not to use Occlumency lessons as an excuse to give his godson a hard time but was told off. "But surely you have noticed that Potter is very like his father?" said Snape, adding that the boy was arrogant like James Potter, and that criticism bounced off him. The two adults pulled out their wands but thankfully the Weasleys arrived just in time and they stopped.
Harry's first Occlumency lesson took place in chapter 24 of the fifth Harry Potter book at Snape's office. When Harry asked why he needed Occlumency, Snape replied it was because the Dark Lord was skilled at Legilimency (the ability to extract feelings and memories from another person's mind) and only those skilled at its defense - Occlumency would be able to close their minds to him. As brutal and challenging as these lessons were, they provided Harry with an opportunity to discover the secrets of Snape's past. He found out that Snape's opinion of his father wasn't all that misplaced. James Potter was, in fact, arrogant, and he made Snape's life at Hogwarts miserable.
Snape's Worst Memory
In chapter 28 of Harry Potter and the Order of the Phoenix, "Snape's Worst Memory," Harry, in Snape's absence, gave the contents of his Pensieve a prod and dove into his memories. He found that there was so much more to this First Wizarding War veteran of a father. Harry stood in the middle of the Great Hall and watched Snape and his father take their Defense Against the Dark Arts - Ordinary Wizarding Level test. They were about fifteen or sixteen, around his age, and therefore in their fifth year.
Harry also saw James' best friends, i.e., the three mischief-making Gryffindors - the Marauders. James rumpled up his hair during the test, and Harry noticed his father often did so to keep it from looking too tidy. The exam concluded and Harry saw the Marauders loaf away in the shade of a beech tree on the edge of a lake. Snape who was engrossed in the OWL paper also sat nearby. As the memory progressed, Harry saw James and Sirius approach Snape. "All right, Snivellus?" said James and that's where all trouble began.
Snape who was caught off guard, reached for his wand, but James disarmed him and hexed his movement using "Impedimenta!" The students all around gathered and while some were entertained, others, such as Harry's mother, Lily Evans felt disgusted. Snape couldn't defend himself because his wand was ten feet away, so Lily stepped in and asked James to leave her friend alone. She called James "an arrogant, bullying toerag" which explained all that hair rumpling. "Snape's Worst Memory" from Harry Potter and the Order of the Phoenix also reveals that Remus Lupin found James' actions unentertaining, but Sirius and Peter Pettigrew laughed out loud. James declared he'd spare "Snivelly" if Lily were to go out with him, but she refused. In the meantime, Snape found the chance to pick up his fallen wand and lay it on James. He directed a hex at James that gashed his face and in retaliation, was lifted upside-down.
James finally did leave Snape alone and called him lucky because Lily came to his aid. What followed next changed the course of Snape and Lily's friendship. Snape, in a fit of rage, declared he didn't need help from "filthy little Mudbloods" like Lily, and this statement enraged her. She delivered an earful about how James was nothing but a big show-off, a bully, and just as bad as Snape and left. James once again hoisted Snape, this time threatening to take off his pants. Harry never found out what happened next as Snape walked into his dungeon and caught him viewing his memories.
The Origins Of The Rivalry
Chapter 33 of Harry Potter and the Deathly Hallows "The Prince's Tale," offers more insight into James and Snape's rivalry. In Harry Potter and the Deathly Hallows: Part 2, Harry poured Snape's memories into the Pensieve and discovered the origins of this rivalry. The megamix of Snape's memories from the book and the movie explain why he turned out the way he did. Snape knew Lily from before Hogwarts, and they were the best of friends. He was a kid from the wrong side of the tracks and a broken family who enjoyed a local muggle-born witch, Lily Evans' company.
As explained in the book, the pair met James Potter and Sirius Black on Hogwarts Express and that's precisely where the rivalry began. The book also explains that Lily hated some of the people Snape hung out with because they used Dark Magic. Furthermore, as the memories progress, and the scenes reform, Harry in the book learns that Sirius pulled a prank and lured Snape down the tunnel beneath the Whomping Willow and James saved him from some kind of harm down there. The truth is, this tunnel led to the Shrieking Shack where Remus Lupin lay in his transformed Werewolf state. Lily and Snape argued over whether James played the hero, and the latter maintained he was merely saving his and his friends' necks.
Finally, Snape's words, "James Potter fancies you!" provide deeper context into the rivalry. The rich and famous James had everything that his foe didn't. He also fancied the one girl Snape had loved since forever. While James had her, Snape lived with unrequited feelings. More importantly, besides the day-to-day clique politics at Hogwarts, James and Snape fought on the opposite side of the Wizarding War. Years later, the reformed Snape came to resent Harry because the boy reminded him of his foe from the yesteryears. "—mediocre, arrogant as his father, a determined rule-breaker, delighted to find himself famous, attention-seeking and impertinent—" is how he described the young wizard in the fifth Harry Potter book. Truth is, Snape was partially right all his adjectives described James Potter before he matured into the man Lily Evans fell in love with.