The nick of a branch is a short name given to that branch.
Nicks are useful when working on different branches all belonging to the same project. For example one could have a master branch called reader and additional branches with nicks like reader-optimise-speed, reader-drop-old-dependencies, reader-use-starttls, each one containing few changes focused on a precise feature.
Nicks are preserved in clone operations. For example, if the branch lp:reader-fix-bug-12 has the nick reader-fix-bug-12 and that branch is cloned with
bzr clone lp:reader-fix-bug-12 test-fix-12
the branch will be cloned in a directory named test-fix-12, but it will still have the nick reader-fix-bug-12