This television prop is a "tribble," a fictional alien creature featured in the episode "The Trouble With Tribbles" in the "Star Trek" TV show, which aired on NBC from 1966-1969. A tribble was a living furry creature on Space Station K-7. They did nothing but eat and multiply. Eventually, tribbles are found aboard the Spaceship Enterprise and created an enormous problem by over-multiplying.
The popularity of this episode led the writer, David Gerrold, to produce the book of the same title, "The Trouble With Tribbles," published by Ballantyne as a paperback in 1973, in which tribbles threaten to overun the entire galaxy. The episode also led to a variety of tribble merchandise and their own cult following almost as big as the show itself.
Paramount gave this prop to the Museum in 1973.
This object is on display in Kenneth C. Griffin Exploring the Planets Gallery at the National Air and Space Museum in Washington, DC.