The traditional first-year economics minimum wage story starts with a minimum wage that is binding above the equilibrium wage. This increases the quantity of labour supplied (more people wanting to work more hours at the higher wage), and decreases the quantity of labour demanded (fewer employers willing to employ workers for fewer total hours), leading to excess supply of labour (unemployment). Further increases in the minimum wage simply exacerbate the problem. And the early empirical research supported that story.
Then came the research that started to debunk the idea of minimum wages leading to increased unemployment, mostly involving cross-border pairs of counties in the U.S., and often focusing on teenagers or fast food workers (e.g. probably the most famous piece is this one by David Card and Alan Krueger (ungated)). In support of these findings, the argument was made that employers had monopsony power, and minimum wages would therefore increase employment (see here for an explanation of this).
Last week Adam Ozimek updated the state of research on the minimum wage. To briefly summarise Adam's summary: several recent papers, improving on the methodology of earlier contributions, have found significant job losses when the minimum wage is increased, "the lost job experience hurt workers even after they eventually found new jobs", and that "low-skilled workers move out of states that raise minimum wages" (which likely means that many research projects using macro data will run the risk of underestimating the disemployment effects of minimum wage increases).
So, those of us still teaching that high minimum wages increase unemployment can breathe a little easier, with the latest research again providing support for the disemployment effects of higher minimum wages.
[HT]: Eric Crampton at Offsetting Behaviour