An involuntary part-time worker is a worker who would like full-time work but can't find it. An involuntary part-time worker is also called a "part-time worker due to economic reasons".
There are two major reasons why somebody would be an involuntary part-time worker:
1) Their hours have been cut from their previously full-time job
2) They can not find a full-time job despite wanting one
It's interesting to note that an "involuntary part-time worker" is counting as being employed, even if they are only working two hours per week despite wanting full-time unemployment. So, if you are working 2 hours per week at McDonald's but have a family to feed, the Bureau of Labor Statistics still counts you as being employed. This is the reason why the "official" unemployment rate can be so far removed from reality.
The United States currently has a distressingly high number of involuntary part-time workers due to the economic malaise that is currently about to stretch into its sixth year.