There are a bunch of different ways of measuring unemployment. The two you here most commonly about are the regular unemployment rate—the share of people who tell a BLS survey that they are actively looking for a job but don't regularly have one—and the so-called U6 rate which adds in people who've stopped looking for work because they couldn't find a job ("marginally attached workers") and people who are working part time but say they'd like to find a full-time job ("part-time for economic reasons"). As Paul Krugman wrote over the weekend, neither of these measures is the "real" unemployment rate. The economic concept of unemployment is simply a bit ambiguous in a real world context. And in practice, he says, "these measures tell pretty much the same story about the ups and downs of the labor market" in the sense that U-6 never goes up while regular unemployment goes down.
On the other hand, not only are U6 and unemployment both strongly cyclical but the gap between them is strongly cyclical:
I'm not exactly sure why that is, but it looks like in a quantitative sense the main issue is that recessions lead to a surge in the number of workers who are part-time for economic reasons:
This is important in terms of tedious debates about skills-mismatch and structural blah blah blah because whatever you think about the completely jobless it's clearly not the case that people who are working part-time lack the skills to be gainfully employed. They are gainfully employed! If you're working at the Wendy's 25 hours a week and tell your boss you want more shifts and your boss tells you that he doesn't have any more shifts to hand out, that's the very definition of slack demand conditions. If more customers started showing up at the restaurant then it would be profitable to employ more person-hours of labor and the members of the existing staff who say they want more hours would be a natural place to turn.