Yeah, there are a few problems with minimum wage, that I can see/know of. And a few more with wages in general, especially CEO-level ones. But first a minor disclaimer: I'm right-winged, hardcore Laissez-Faire proponent; I don't live in USA; I have some economical knowledge, but it is not gained during formal studies (but I was using university-level materials to gain it).
So, I'm from Poland, as some of you might or might not know. As we are going through elections right now, the minimum wage is (of course!) being discussed here, too. If mix something up in regards to American regulations, please, let me know. Our laws are probably somewhat different than American (*) so I might confuse something.
The main problem we need to address is that pointed out to HugoLuman: with increased minimum wage, there is a risk of lay-offs. Not only in small companies that won't be able to afford to increase wages, although that will surely happen too; there are some things that are barely profitable with minimum wage of 7.25$, and they will just stop working after an increase. But think also of big companies. If there are only minimum wage workers, and the wage is increased by 10%, the management might try to get away with letting 10% of the crew go and just working the rest harder, since they are already getting 10% more. They will cut hours in fast food joints and the hamburgers that fallen on the floor will lay there until rush hour ends and somebody is free to pick them up.
In general: those with jobs will earn more, but there will be less people with jobs.
Work is a resource, like everything else. Let's say cheap milk costs 1$ a gallon, and expensive brand costs 1.5$ per gallon, and there are people buying a gallon a week of cheap one. And suddenly there is a minimum price of milk: 1.25$ a gallon. Some people will start paying 1.25$, but others won't, they will go back to drinking water barbarians. And some will think 'well, since the cheap one is almost as expensive as the expensive one, I will just buy the expensive one'. But generally, there will be less milk sold, because nobody will start to think 'since the milk is more expensive, I will now buy more milk!'...
For starters, our minimum wage is monthly, not hourly; but much more importantly, our minimum wage is tied to a number of other values declared by law, like unemployment benefits, various social payments (for disabled or other), level of taxes paid by some people (not only minimum-wage earners, but also self-employed, if I'm not messing anything) etc. This means that changing our minimum wage has much bigger repercussions that one might think.