Most stories of Hades and Persephone are the same. That Hades saw Persephone and fell in love with her, so he asked her father- his brother if he could have her hand in marriage. Zeus agreed, but advised Hades to kidnap her as Demeter -Persephone's mother- would never agree to it.
So Hades waited for Persephone to be alone, and then he rode up in his chariot, grabbed the girl and took her back down to the Underworld. She eventually grew to like it, but her mother threw the tantrum of the century, and they had to come up with a compromise, where she would spend a third of the year with her mother, a third with Hades and a third with whoever she wanted.
But, that isn't the original story, the original is so much better, the one of Persephone getting kidnapped is terrible and I hate that people don't know the real story.
Her name was actually Kore in the beginning, which meant 'little girl', we'll get to when it changes later.
So, Kore was walking around one day, picking flowers or whatever, when she stumbles upon an entrance to the Underworld, and decides to explore it, because why not? She walks to the Underworld, and decides that she likes it here.
Hades did not bring her here, he was confused as to why she was there considering she wasn't dead, and so wasn't supposed to be there. Until then, he was the only alive person in the Underworld.
Kore decided she liked the place so much that she was going to stay there, and Hades just agreed to it. He probably had started to fall for her by then. But then, Demeter, mother of the year, threw a bitch fit, and then Zeus had to intervene.
Kore really loved the underworld, and she would love Hades later on when they got married, but her mother was being dramatic, and that lead to Zeus having to say some things. He eventually just gave up and agreed to Kore staying in the Underworld as long as she visited her mother, and he also changed her name from Kore, to Persephone, which meant 'the one who brings chaos'.