Harry tries to prevent Quirrell from stealing the Sorcerer’s Stone, and he uses the Mirror of Erised to do it.
One of the major conflicts in this first book of the series is a character vs. character conflict between Harry and Quirrell. A character vs. character conflict is one in which a character has a struggle between an opposing force represented as another character. A conflict is resolved when the source of the initial conflict has been dealt with in some way. The resolution of the conflict was that Harry defeated Quirrell when Quirrell could not find the Sorcerer's Stone.
Harry Potter is a wizard who attends a special school for wizards. He is convinced that one of his teachers is trying to kill him. He is correct about that, partially. Unfortunately, for most of his first year he is focused on the wrong one and the wrong reason.
Harry discovers that the school is hiding a special jewel called the Sorcerer’s Stone, made by an alchemist named Nicholas Flamel, which “will make the drinker immortal” (Ch. 13). It is being guarded in a secret part of the castle.
Harry heard two of his teachers, Snape and Quirrel, discussing the Stone. Since Quirrell was a sniveling wimp and Snape was overbearing and seemed to not like Harry, Harry assumed that Snape was the enemy. He had also seen Snape with scratches on his leg, and assumed that he had tangled with the Three-headed dog guarding the Stone ("Fluffy").
"Blasted thing," Snape was saying. "How are you supposed to keep your eyes on all three heads at once?" (Ch. 11)
It does look suspicious. Snape is also plenty mad when he sees Harry. You can see how Harry would come to the conclusion that Snape is the one trying to get the Stone. Harry and Hermione also think that Snape is trying to curse Harry during a Quidditch match, but Snape is trying to perform a counter-curse (Ch. 17).
Harry and his two friends, Ron and Hermione, had to undergo a series of tests to get to the stone. These included a killer plant, a chess match, a Quidditch test with flying keys, and a potion riddle, before Harry finally got to the end.
Once reaching the end, he found himself faced with Professor Quirrell and the Mirror of Erised. This mirror was something that Harry was familiar with. He knew that it showed a person what he desired most. (“Erised” is “desire” spelled backwards, because the mirror twists our desire).
Quirrell had passed all of the other tests, but was flummoxed by this one. It turned out that the evil Dark Wizard Lord Voldemort was using him for a host.
Quirrell came back out from behind the mirror and stared hungrily into it.
"I see the Stone... I'm presenting it to my master... but where is it?" (Ch. 17)
Since Quirrell does not have an innocent motive, he will never get the Stone. This was Dumbledore, the Headmaster’s plan. The mirror is the final test. Harry has it and is able to keep it safe. Voldemort and Quirrell can’t get him, or the Stone. Dumbledore and the others are able to come and rescue Harry after he passes out. Dumbledore explains to him when he wakes up that only someone who wanted to find the Stone and protect it, but not use it, would be able to get it. He tells him the stone has been destroyed.
This is a conflict that did not get entirely resolved. It was not really Quirrell that Harry was battling, it was Voldemort. Harry will have to face him again and again, because he is the determined to rule the world, and Harry is determined to stop him.