When a plane goes into a tailspin, it rotates as it dives toward the ground. While pilots sometimes go into a deliberate tailspin, an unexpected tailspin would be terrifying.
Because of the dramatic nature of an aircraft's tailspin, the word is also used figuratively to mean "a chaotic or panicked situation or state." Getting terrible news sends some people into a tailspin, while others are able to somehow stay calm. The word dates from about 1915, and the figurative meaning first appeared in print in the 1930s.