Meaning 1
The back or "tail" of the plane drops and starts spinning
Sentence 1
The plane went into a tailspin before the pilot bailed out.
Meaning 2
to drop or fall rapidly ( as above in "nosedive"); to be anxious or extremely concerned
Sentence 2
When I lost my job, my budget went into a tailspin. He went into a tailspin when his secretary said she was quitting.