In the autumn of 1959, shy Todd Anderson begins his senior high school at Welton Academy, an all-male, elite prep boarding school. He is assigned one of Welton's most promising students, Neil Perry, as his roommate and is quickly accepted by Neil's friends: romantic Knox Overstreet, overachiever Richard Cameron, best friends Steven Meeks and Gerard Pitts, and mischievous beatnik Charlie Dalton.
On the first day of classes, they are surprised by the unorthodox teaching methods of new English teacher John Keating, a Welton alumnus who encourages his students to "make your lives extraordinary", a sentiment he summarizes with the Latin expression carpe diem ("seize the day"). Subsequent lessons include having them take turns standing on his desk to teach the boys how they must look at life in a different way, telling them to rip out the introduction of their poetry books which explains a mathematical formula used for rating poetry, and inviting them to make up their own style of walking in a courtyard to encourage them to be individuals. His methods attract the attention of strict headmaster Gale Nolan.
Upon learning that Keating was a member of the unsanctioned Dead Poets Society while he was at Welton, Neil restarts the club and he and his friends sneak off campus to a cave where they read poetry and verse, including their own compositions. As the school year progresses, Keating's lessons and their involvement with the club encourage them to live their lives on their own terms. Knox pursues Chris Noel, a girl who is dating a football player and whose family is friends with his. Neil discovers his love of acting and gets the lead in a local production of A Midsummer Night's Dream, despite the fact that his domineering father wants him to go to medical school. Keating helps Todd come out of his shell and realize his potential when he takes him through an exercise in self-expression, resulting in his composing a poem spontaneously in front of the class.
Charlie publishes an article in the school newspaper in the name of the Dead Poets Society demanding that girls be admitted to Welton. Nolan uses corporal punishment to coerce Charlie into revealing who else is in the Dead Poets Society, but he resists. Nolan also speaks with Keating, warning him that he should discourage his students from questioning authority.
Neil's father discovers Neil's involvement in the play and tells him to quit on the eve of the opening performance. Devastated, Neil goes to Keating, who advises him to stand his ground and prove to his father that his love of acting is something he takes seriously. Neil's father unexpectedly shows up at the performance. He takes Neil home and tells him he is forcing him into military school. Unable to find the courage to stand up to his father, a distraught Neil commits suicide.
Nolan investigates Neil's death at the request of the Perry family. Richard blames Neil's death on Keating to escape punishment for his own participation in the Dead Poets Society, and names the other members. Confronted by Charlie, Richard urges the rest of them to let Keating take the fall. Charlie punches Richard and is expelled. Each of the boys is called to Nolan's office to sign a letter attesting to the truth of Richard's allegations, even though they know they are false. When Todd's turn comes, he is reluctant to sign, but does so after seeing that the others have complied.
Keating is fired and Nolan takes over teaching the class. Keating interrupts the class to collect personal articles; before he leaves Todd shouts that all of them were forced to sign the letter that resulted in his dismissal and that Neil's death was not his fault. Todd stands on his desk and salutes Keating with the words "O Captain! My Captain!". Over half the rest of the class does the same, ignoring Nolan's orders to sit down. Keating is deeply touched by their gesture and realizes his teaching has made a lasting impact. He thanks the boys and departs.

Question: How does Keating instruct his students on how to live their life?
Answer: He inspires them to be individuals and to seize the day.
Alternative answer: By quoting "carpe diem"