Rin is an orphan who stumbles upon Sesshomaru while he's recovering from a fight with InuYasha. Traumatized by watching bandits slaughter her family, she'd become mute and was basically surviving on her own, tolerated but hardly welcomed by a nearby village.
She starts bringing Sesshomaru food and water without being asked, though he mostly ignores her efforts. The one time he casually asks about the injuries she got from villagers, she lights up in a way that actually confuses him. When Sesshomaru finally leaves, he catches her scent on the wind—it's blood. He finds her killed by wolves, and something about her death moves him (whether it's genuine sympathy, curiosity about his sword Tenseiga's power, or both is left ambiguous). He uses Tenseiga to bring her back.
After that, Rin follows him everywhere with an intensity that drives Jaken crazy, and she's regained the ability to speak. Sesshomaru has basically become her protector, stepping in whenever she's actually in danger. Her name literally means "cold."
Content compiled by AnimeList.moe from publicly available sources.