Voluntary versus Involuntary Unemployment
At a very basic level, unemployment can be broken down into voluntary unemployment- unemployment due to people willingly leaving previous jobs and and now looking for new ones- and involuntary unemployment- unemployment due to people getting laid off or fired from their previous jobs and needing to find work elsewhere. Not surprisingly, economists generally view involuntary unemployment as a larger problem than voluntary unemployment since voluntary unemployment likely reflects utility-maximizing household choices.
The easiest type of unemployment to explain is known as frictional unemployment. Frictional unemployment is unemployment that occurs because it takes workers some time to move from one job to another. While it may be the case that some workers find new jobs before they leave their old ones, a lot of workers leave or lose their jobs before they have other work lined up. In these cases, a worker must look around for a job that it is a good fit for her, and this process takes some time. During this time, the individual is considered to be unemployed, but unemployment due to frictional unemployment is usually thought to last only short periods of time and not be specifically problematic from an economic standpoint. This is particularly true now that technology is helping both workers and companies make the job search process more efficient.
Frictional unemployment can also occur when students move into the work force for the first time, when an individual moves to a new city and needs to find work, and when women re-enter the work force after having children. (Note in the last case, however, that maternity leave doesn't count as unemployment!)
It's probably not surprising that unemployment is higher during recessions and depressions and lower during periods of high economic growth. Because of this, economists have coined the term cyclical unemployment to describe the unemployment associated with business cycles occurring in the economy. Cyclical unemployment occurs during recessions because, when demand for goods and services in an economy falls, some companies respond by cutting production and laying off workers rather than by reducing wages and prices. (Wages and prices of this sort are referred to as "sticky.") When this happens, there are more workers in an economy than there are available jobs, and unemployment must result.
As an economy recovers from a recession or depression, cyclical unemployment tends to naturally disappear. As a result, economists usually focus on addressing the root causes of the economic downturns themselves rather than think directly about how to correct cyclical unemployment in and of itself.
There are two ways to think about structural unemployment. One way is that structural unemployment occurs because some labor markets have more workers than there are jobs available, and for some reason wages don't decrease to bring the markets into equilibrium. Another way to think about structural unemployment is that structural unemployment results when workers possess skills that aren't in high demand in the marketplace and lack skills that are in high demand. In other words, structural unemployment results when there is a mismatch with workers' skills and employers' needs. Structural unemployment is thought to be a pretty significant problem, mainly because structural unemployment tends to be largely of the long-term variety and retraining workers is not a cheap or easy task.
Seasonal unemployment is, not surprisingly, unemployment that occurs because the demand for some workers varies widely over the course of the year. (Pool lifeguards, for example, probably experience a decent amount of seasonal unemployment.) Seasonal unemployment can be thought of as a form of structural unemployment, mainly because the skills of the seasonal employees are not needed in certain labor markets for at least some part of the year. That said, seasonal unemployment is viewed as less problematic than regular structural unemployment, mainly because the demand for seasonal skills hasn't gone away forever and resurfaces in a fairly predictable pattern.
(See here for the statistical measures of unemployment used by the Bureau of Labor Statistics.)