The minimum wage is the best example of an issue on which our current discussion as citizens is impoverished by a lack of knowledge of economics. In order to understand the costs and benefits of minimum wage laws, it is necessary to understand supply, demand, and how a market sets prices.
One of the things one learns in Econ 101 is that the market price for a particular good or service is the price that maximizes overall utility (i.e., happiness). Before we consider the price of labor, let's just consider the price of umbrellas as an example. In any particular market, there is going to be a particular demand for umbrellas at a particular price (for example, let's say that 10,000 people would be willing to buy umbrellas today if they cost $10 a piece but say, only 7,000 people would be willing to buy umbrellas if they cost $15 a piece). Similarly, there is going to be a particular supply of umbrellas at a particular price (say, people would be willing to produce 12,000 umbrellas if they could sell them for $20 a piece but are only willing to produce 6,000 umbrellas if they can only sell them for $12 a piece). You can draw a graph that shows two curves: a demand curve showing the number of umbrellas people will buy at a particular price (for the demand curve, there will be greater demand the lower the price) and a supply curve showing the number of umbrellas people will produce at a particular price (for the supply curve, there will be greater supply the higher the price). The place where the demand curve and the supply curve cross is the point where the market will set the price for umbrellas. One of the nice properties of the market price is that it maximizes the total happiness to both consumers and producers, and so the overall amount of happiness with the number and price of umbrellas is maximized when we let the market set the price.
OK, now let's consider the minimum wage. Just as with umbrellas, there is a demand curve for unskilled labor (i.e., as the price for unskilled labor goes up, demand for unskilled labor will go down) and a supply curve for unskilled labor (i.e., as the price for unskilled labor goes up, the number of people willing to supply unskilled labor will go up). The market, left to its own devices, will set the price of unskilled labor at the point where the supply curve crosses the demand curve, and just as in the market for umbrellas, the market price for unskilled labor is the price that maximizes the overall amount of happiness with the price and quantity of unskilled labor. When a minimum wage is introduced, it means that the government is artificially placing the price of unskilled labor above the point where the market would set it. Placing the price above the market price does make some people happier than they would be at the market price: namely, those people who are employed at the new, higher wage. However, as is inevitable whenever you set the price of anything above the point where the market would set it, there are going to be more people hurt by the new wage than are helped. The people who are hurt include 1) consumers who will have to pay higher prices (and higher prices hit the poor much harder than the rich), 2) business owners who employ unskilled labor and had thin profit margins before the wage increase who will be forced out of business because of the price change, and 3) (and most importantly) when you artificially increase the price of unskilled labor, the inevitable result is that there will be fewer such jobs (demand always decreases when you raise the price of anything, including unskilled labor) meaning that many poor people will be thrown out of work by the wage increase. To summarize then, the minimum wage is the nasty business of raising some poor people's wages by throwing other poor people out of work (with higher prices for consumers and more failed businesses thrown in for good measure). The minimum wage is politically expedient because the people who receive the new wage realize that the minimum wage increase made their lives better and will reward the politicians who passed the law that gave them the new wage, but the people who are hurt by the minimum wage do not realize that the reason they don't have a job and are paying higher prices is because of the minimum wage. The minimum wage is thus a way for politicians to buy people's votes at the cost of making almost everybody worse off.
The cases in United States history that are often cited as showing that a minimum wage is necessary are a) the existence of sweatshops in East Coast cities that paid extremely low wages during the period around 1900 when many immigrants were coming to those cities from Europe, and 2) the people described in John Steinbeck's book "The Grapes of Wrath" who migrated from the Midwest to California during the depression looking for jobs picking fruit and found that the fruit picking jobs paid starvation wages. What these two cases have in common is that, in both cases, there were large numbers of people who had exactly the same set of skills who migrated to the same place at the same time. One of the worst things that can happen to human beings is to migrate to a place where there are large numbers of people who have exactly the same set of skills because under those circumstances the supply of those skills is going to be huge while the demand for such skills is going to be much smaller meaning that the price people are going to be willing to pay for those skills is going to be very, very small (too small for someone to support themselves). I would agree that under this limited set of circumstances (specifically, circumstances in which a huge immigrant population with the same set of skills has moved into a geographic area in a very short period of time) the market left to its own devices will not be able to handle the situation because the market's solution (more business enter the area to provide employment) will take too long, and all the workers will have starved by the time the market solves the problem. Government or private charity may be necessary under those circumstance. However, the way the government should intervene is definitely not by imposing a minimum wage. The reason a minimum wage is a bad solution to the problem is that only a few people will be able to get the minimum wage, and the rest will simply be out of work. A much better solution to the problem of a market that has an excess supply of unskilled labor is to move some of the people to an area of the country (or the world) where there is not an excess supply of unskilled labor so that they may make a higher wage.
So, to summarize, the minimum wage is a policy that is politically expedient, but makes the world a less happy place than letting the market set the price of labor. In the few circumstance where the market price of labor would be so low that people would not be able to afford the necessities of life on it, the solution is not to impose a minimum wage, but rather to move people to a place where the supply of unskilled labor is lower.
If you had difficulty following my discussion of the effects of minimum wage laws, here are a few links that may make it clearer. Here is a discussion of the minimum wage by another libertarian who makes arguments similar to mine, but in a slightly different way. Walter Williams, an economist at George Mason University, has several articles about the problems caused by minimum wage laws that can be found here, here, and here. Here is a good discussion by Henry Hazlitt of the effects of minimum wage laws.
Here is Nobel Prize Winner Milton Friedman discussing the terrible effects of the minimum wage:
Here is a really good discussion of supply, demand, and the minimum wage by an economist:
Walter Williams, an economist, explains the problems with the minimum wage: