Q:

Is ';;' a chaining operator in Unix? How does it work?

My teacher lists the following chaining operators:
& && ( ) ; ;; | || new line

How is ;; a chaining operator and what does it do?

A:

Your teacher will probably talk later on about case statements in bash and how each option statement is closed by ;;.
As in:
case $space in
[1-6]*)
  Message="All is quiet."
  ;;
[7-8]*)
  Message="Start thinking about cleaning out some stuff.  There's a partition that is $space % full."
  ;;
9[1-8])
  Message="Better hurry with that new disk...  One partition is $space % full."
  ;;
99)
  Message="I'm drowning here!  There's a partition at $space %!"
  ;;
*)
  Message="I seem to be running with an nonexistent amount of disk space..."
  ;;
esac

See TLDP Bash Guide for Beginners - 7.3. Using case statements

