Note: The literal meaning of something means that the reader is looking at exactly how this thing is being used in the plot. While the figurative meaning of something is looking at deeper meanings within objects or ideas in a work of fiction.
J.D. Salinger uses the carousel and the ring literally to wrap up the novel and display how Holden enjoys watching his sister go round and round on this ride meant for children. In the Natural History Museum with the mummies, Salinger demonstrates how Holden is slowly losing grip on his reality.
The figurative meanings behind these events are much more interesting. All of the objects stated above have to do with the novel's primary theme, which is Holden's fear of growing up.
A carousel is a safe ride for children in that it only goes around in circles. The fake animals on the carousel may move up and down, but it is still a very safe ride. The safety of this ride makes Holden happy. Phoebe, however, starts taking a risk by reaching for that golden ring. This is symbolic of how Holden wishes he could reach for that ring, but is unable to do so.
The museum and the mummies are also symbolic of Holden's insistence of keeping things as they are. The museum hasn't changed since Holden was a child and the mummies are preserved in their wrappings, much the same way he wishes he could preserve Phoebe.