Emily is the girl who owns Jessie before abandoning her in Toy Story 2.
Jessie's memories are shown during the song “When She Loved Me.” As a child, Emily plays with Jessie and treats her as a beloved companion. Over time, Emily grows older and begins to outgrow the toy, eventually leaving Jessie in a donation box.
The flashback explains Jessie's fear of being abandoned and gives emotional context to her resistance to returning to Andy. It also parallels Woody's concern about whether Andy will eventually stop playing with him.
Emily is not one of the children who own other important toys in the series. Bonnie receives many of Andy's toys at the end of Toy Story 3, while Molly is Andy's younger sister and does not own Jessie in the flashback.