Discouraged workers are people of working age who would like a job but have stopped searching, usually because they think no suitable work is available. Because official surveys only count people who are actively looking, they drop out of the unemployment figures and are classed as economically inactive. The full explanation is on the Economic Inactivity page.
Statisticians publish wider measures that add discouraged workers back in, such as the US Bureau of Labor Statistics' U-4 to U-6 rates, because a falling jobless rate can hide people giving up rather than finding work. Long spells out of work can turn into structural unemployment.