In chapter 28, Mankiw focuses on the natural rate of unemployment (a long-run problem). Unemployment is important because it decreases from the economy's GDP, and thus the economy's standard of living. Unemployment is an imperfect measurement from the BLS of those who want to work but who do not have jobs, perhaps because of either frictional or structural unemployment. For example, discouraged workers (individuals who would like to work but have given up looking for a job) are not included in the measurement.
I am a little confused about the "seemingly contradictory" result: most spells of unemployment are short, and most unemployment observed at any given time is long-term; and the example of the four government employment workers did not help.
It is important to note that the unemployment is never zero, but fluctuates around the natural rate of unemployment. This is due to frictional unemployment (it takes time for workers to search for the jobs that best suit their tastes and skills) and structural unemployment (unemployment that results because not enough job positions).