This question has already been widely discussed and answered on Skeptics. For information I will copy the most upvoted and accepted answer, courtesy of Borror0, here:
The answer to this question is "we're not sure yet."
The "old minimum wage research" shows that there is a negative impact, but more recent research - e.g. Card and Kreuger (1993) and Dube, Lester, and Reich (2010) - show that there is no significant effect.
There have been attempts at reconciliation the literature on the subject, but, to the best of my knowledge, nothing resembling a consensus has been achieved amongst economists.
When reviewing the possibility to raise the minimum wage again, Québec's Interdepartmental Committee for the Review of the Minimum Wage compiled a short review of the literature. It's written in language most people will understand, and only six pages long. It's definitively worth reading.
In the event you don't feel like reading it, the most interesting passage is the following:
Economic debate concerning the minimum wage has essentially been focussed around two
subjects of discussion. These are the impacts that such a policy has on employment levels and its effects on the distribution of wealth. Economic theory generally approaches the impacts that the minimum wage may have on employment using mainly two models. These are a “pure and perfect” competition model (or neoclassical) and an imperfect model called “monopsony”.
Over the last 40 years, economic studies based on the neoclassical model show that mainly young people less than 24 years old are generally the most affected by job reductions that are likely to take place when the minimum wage increases. Elasticity calculated by these models varies between –0.1 and –0.3, meaning that a 10% increase in the actual minimum wage will generate a reduction of employment for young people varying between 1% and 3%. An econometric made by the Department of Finance leads to a similar conclusion for young people aged between 15 and 19.
However, numerous conditions must be met for this model to apply, and because of that, many economists have challenged its relevancy, especially since the nineties. These economists prefer the use of a monopsony model that includes market imperfections. Results obtained using this model are very different from those obtained with the neoclassical model. In fact, they lead to the conclusion that the increases in the minimum wage that occurred over the last 15 years in certain areas of North America and Europe did not hinder employment.