To understand the above graph is to understand why the minimum wage is harmful. The line in blue is the supply curve for labor, indicating that when wages are higher, the supply of labor increases as more people are willing to enter the workforce or work more hours. The line in red is the demand curve for labor; as workers get more expensive, fewer people are willing to hire workers. The point where supply and demand intersect is the market clearing point, to which quantity of workers and wage levels will tend toward naturally. Wages at the market clearing point are w*, with quantity of q*.
The horizontal line labeled "minimum wage" shows what happens if you set a price floor on labor that is higher than the market clearing level. More people start looking for work (indicated by qS) and fewer people are looking to hire them (indicated by qD). The difference between the quantity of labor supplied and the quantity of labor demanded is an indicator of unemployment or underemployment. The higher that line is, the more unemployment is caused by it.
People who were making more than the new minimum wage will not be affected, so it's important to look at what the impact will be on people who were making less than the new legal minimum. Some of them will get raises (up to the new minimum), and some of them will lose their jobs. That's it for the economic effects of the minimum wage, but there are also social effects and political effects as well.
As mentioned before, the supply of workers increases when the minimum wage is raised above the market clearing level. If employers have any sort of preference for workers on the basis of age or race (and even though it's technically illegal, a white 30-something is probably going to have an easier time getting hired than a black teenager) then expanding the quantity of labor supplied while reducing the quantity of labor demanded means that a disproportionate number of the people who lose their jobs will be members of those less-preferred groups; mostly teenagers, and to a lesser extent ethnic minorities. The Bureau of Labor Statistics indicates that as of January 2013, the seasonally adjusted unemployment rate of various groups was as follows:
White males, 20 years and up: 6.6%
White females, 20 years and up: 6.4%
Whites (both sexes), 16-19 years: 20.8%
Black males, 20 years and up: 13.4%
Black females, 20 years and up: 12.3%
Blacks (both sexes), 16-19 years: 37.8%
Another report on characteristics of minimum wage workers for 2011 (the most recent year available) shows that the minimum wage affects teenagers disproportionately. While blacks (at 6%) are slightly more likely to work for minimum wage or less than whites (at 5%), workers aged 16-19 earned minimum wage or less 22.8% of the time. At ages 20-24, the rate of minimum wage workers drops to 3.8%, which is still nearly three times higher than the 1.3% of the over-25 workforce that makes minimum wage or less. All groups considered, the under-25 cohort comprises about half of all workers making minimum wage or less. While Obama talked about how hard it is for a couple with two children to get by on minimum wage, fewer than one percent of married couples earn the minimum wage. It is simply a political distraction away from the people who actually make minimum wage.
The reason that the minimum wage persists despite being awful economic policy is because it is a political no-lose situation for the party that has set itself up on the principles of class warfare. The worker who gets a pay raise as a result of an increase in the minimum wage is grateful to the government for raising the minimum, while the worker who loses their job when the minimum wage is increased is far more likely to blame corporate greed, as his political masters tell him to. He's probably never seen intersecting supply and demand curves before in his life and doesn't understand that it is the government's fault that he is now a legally non-profitable (and therefore unemployed) worker. Instead, he goes on the dole, and is grateful to the government for providing support in his unemployment. Businesses who pay the minimum wage will be upset, but the bourgeois are irrelevant in popular elections and most of them don't vote Democrat anyway, so what's there to lose by pissing them off? Minimum wage increases will likely be vocally supported by companies that pay higher than the minimum wage but are competing with companies that do not; Wal-Mart supports a higher minimum wage because they pay more than that already, but smaller grocery stores often don't. For politicians, the minimum wage is a tool to ingratiate themselves to undereducated voters. For big businesses, the minimum wage is a tool to put pressure on smaller competitors. Neither of these groups care as much about the economy as they do about their own success, so they support the minimum wage.