Three years after the destruction of the Death Star, the Rebel Alliance has been driven from their former base on Yavin IV by the Galactic Empire. The Rebels, led by Princess Leia, set up their new base on the ice planet Hoth. The Imperial fleet, led by Darth Vader, continues to hunt for the Rebelsâ new base by dispatching probe droids across the galaxy.
While investigating a potential meteor strike, Luke Skywalker is captured by a wampa. He manages to escape from its cave with his lightsaber, but soon succumbs to the brutally cold temperatures. The ghost of his late mentor, Obi-Wan Kenobi, instructs him to go to the Dagobah system to train under Jedi Master Yoda. He is found by Han Solo, who uses the warmth of his dead tauntaun to keep Luke warm while he sets up a shelter. Han and Luke make it through the night and are rescued by Zev Senesca.
On patrol, Han and Chewbacca discover the meteor Luke had planned to investigate is actually a probe droid, which alerts the Empire to the Rebelsâ location. The Empire launches a large-scale attack, using AT-AT Walkers to capture the base. Han and Leia escape on the Millennium Falcon with C-3PO and Chewbacca, but their hyperdrive malfunctions. They hide in an asteroid field, where Han and Leia grow closer. Vader summons bounty hunters, including the notorious Boba Fett, to assist in finding the Falcon. Luke, meanwhile, escapes with R2-D2 in his X-wing fighter and crash-lands on the swamp planet Dagobah. He meets a diminutive creature who is revealed to be Yoda; after conferring with Obi-Wan's spirit, Yoda reluctantly accepts Luke as his pupil.
After evading the Empire, Han sets a course for Cloud City, a floating colony in the skies of the planet Bespin. Cloud City is run by Han's old friend, Lando Calrissian. Unknowingly, the Millennium Falcon has been tracked by Boba Fett; shortly after they arrive, Lando leads the group into a trap and they are handed over to Darth Vader. Vader plans to use the group as bait to lure out Luke, intending to capture him alive and take him to the Emperor. During his training on Dagobah, Luke sees a premonition of Han and Leia in pain and, against Yoda's wishes, leaves to save them.
Vader goes back on his agreement with Lando and takes Leia and Chewbacca into custody. He intends to hold Luke in suspended animation and, as a test, freezes Han alive in a block of carbonite. Vader hands the frozen Han over to Fett, who intends to leave for Tatooine to deliver Han to Jabba the Hutt and claim the bounty on Solo's head. Lando, who was forced into cooperating with the Empire, initiates an escape and frees Leia and the others. They try to save Han but are unable to stop Fett. They then flee Cloud City in the Falcon.
Luke arrives at Cloud City and falls into Vader's trap. The two engage in a lightsaber duel that leads them over the city's central air shaft where, as his mentors warned, Luke proves to be no match for Vader who severs Luke's right hand, causing him to lose his weapon. After Luke refuses to join Vader against the Emperor, Vader reveals that he is Luke's father. Horrified, Luke falls through the air shaft. He is ejected beneath the floating city and makes a desperate telepathic plea to Leia, who senses it and persuades Lando to return for him in the Falcon. R2-D2 repairs the Falcon's hyperdrive, allowing them to escape the Empire.
Later, aboard a Rebel medical frigate, Luke's amputated hand is replaced with a robotic prosthetic. Lando and Chewbacca set off for Tatooine in the Falcon in order to save Han. As the Falcon departs, Luke, Leia, R2-D2, and C-3PO look on and await word from Lando.

Question: How is Yoda convinced to take on Luke Skywalker as a student?
Answer: The spirit of Obi-Wan Kenobi talks him into it.
Alternative answer: He talks to Obi-Wan's spirit