Frictional unemployment is defined as the unemployment that occurs because of people moving or changing occupations. Demographic change can also play a role in this type of unemployment since young or first-time workers tend to have higher-than-normal turnover rates as they settle into a long-term occupation. An important distinguishing feature of this type of unemployment, unlike the two that follow it, is that it is voluntary on the part of the worker.
Structural unemployment is defined as unemployment arising from technical change such as automation, or from changes in the composition of output due to variations in the types of products people demand. For example, a decline in the demand for typewriters would lead to structurally unemployed workers in the typewriter industry.
Cyclical unemployment is defined as workers losing their jobs due to business cycle fluctuations in output, i.e. the normal up and down movements in the economy as it cycles through booms and recessions over time.In a recession, frictional unemployment tends to drop since people become afraid of quitting the job they have due to the poor chances of finding another one. People that already have another job lined up will still be willing to change jobs, though there will be fewer of them since new jobs are harder to find. However, they aren't counted as part of the unemployed. Thus, the fall in frictional unemployment is mainly due to a fall in people quitting voluntarily before they have another job lined up.
But the drop in frictional unemployment is relatively small and more than offset by increases in cyclical and structural unemployment. One of the big questions right now is whether the US economy is suffering, for the most part, from structural or cyclical unemployment. If it's cyclical, then there's a good chance that government intervention can help. If it's structural, i.e. a decline in automobile production and manufacturing more generally, a decline in home construction, and a decline in the financial industry all of which free workers that need to be absorbed elsewhere in the economy, there's less that can be done and some do not think that government can do much at all about this type of problem (though as I note below, I disagree). Thus, the debate is between those who say our current unemployment problem is largely cyclical and hence we need more government action, and those who say it's structural and hence there's very little that government can do. We will just have to wait for the structural changes to take place, and that takes time.
I don't think this debate can be answered by moving close to the polar extremes and declaring it's mainly a structural or cyclical problem. For me, it seems obvious that part of the problem is structural. The real question is how large the structural component is and what can be done about it. But no matter how large it is -- take a very liberal estimate of the size -- I don't think there's any way to deny that there is a substantial cyclical component on top of it that demands government action. It's true that the size of the government action to offset the the cyclical downturn should be connected to the size of the cyclical unemployment problem, but the problem is big enough that politicians won't come anywhere near to overdoing it. The most optimistic view of what Congress might do would still leave them short of what is needed. We don't know the exact structural-cyclical breakdown, but the cyclical problem is certainly larger than any imaginable Congressional response. So the excuse for inaction based upon the "it's all structural" claim isn't persuasive.
What about the structural problem, does government have any role to play, or does it have to rely upon the private sector to solve this problem by itself? Several points on this. First, even if the problem is mostly structural, the government can still provide people with jobs to bridge the gap until the structural changes are complete. To me, this is better than simply extending unemployment compensation since it allows individuals to contribute something (e.g. work on a project the local community needs). And by giving people jobs, or at least government aid through unemployment compensation, we increase aggregate demand and the that helps firms to do better and speeds the transition.
Second, government can ease the structural problem by making it easier for businesses and individuals to relocate. People are understandably reluctant to leave the place they have lived for years and years, but government incentives to relocate (tax breaks, subsidies, etc.), can help. So can efforts to provide individuals in communities suffering from high unemployment with information about where job prospects are better, as can retraining programs (though these aren't always as effective as hoped). In a deep, widespread recession places that need workers may be hard to find, but not always and knowing where there are better opportunities for employment can be helpful. Businesses can also be induced to relocate through tax and other incentives, though the tax competition that accomplishes this may strip local governments of needed tax revenue, so I'd prefer these programs originate at the federal level. And there can also be government encouragements to speed the investments that are needed to complete the transition.
But the main things I want to emphasize are that no matter how large the structural problem is, cyclical unemployment is also a big problem, so the claim that government is powerless because it's all structural doesn't hold. And the claim that the existence of a structural problem means there's nothing the government can do is also incorrect. If nothing else, the government can help workers during the transition. In addition, though the opportunities here are more limited, there are also things the government can do to make the transition happen sooner rather than later.