The Snow Globe is a prized possession of Coraline's Mother, a souvenir from the zoo. On moving into the Pink Palace, Coraline puts it in front and center on the living room mantelpiece.
When the Other Mother kidnaps Coraline's Real Parents, she imprisons them somewhere in the house. At the climax of the film, Coraline remembers a vision she saw of her parents shivering in a snowstorm, and realizes her parents are inside the Snow Globe. She tricks the Other Mother to move to the other side of the room and unlock the Little Door by pretending to erroneously believe her parents are trapped inside the tunnel, then throws the Cat at the Other Mother to create a distraction so that she can grab the Snow Globe from the mantelpiece.
At the end of the film, it is revealed that the process of freeing the Real Parents shattered the Snow Globe.
Mrs. Jones is quite annoyed rather than grateful, since she has retained no memory of their entrapment and rescue.
In the book, the Jones had own no snow globes. It originally appeared on the drawing rooms mantel piece in Coraline's last trip to the Other World, she took some interest in the object noticing two little people but had not truly much cared about it. When Coraline found the last Ghost Eyes she realizes that is were the Other Mother had been keeping them, after tricking the Other Mother into opening the door and setting the cat on her, Coraline grabbed the snow globe and went back to the real world. When her parents came back, the two little people were gone and it was never mentioned again.