Labor like all other goods and services is controlled by the interaction of Supply and Demand. In short the market demand for something is the ability and desire of everyone in the market to purchase that item at all prices and market supply is the ability and desire of all the businesses in the market to create/sell them at all prices. 
This is a simple graph of just that. 
The place where the supply line and the demand line cross is called equilibrium. If you price a product at equilibrium price (P*) you will succeed in having an equilibrium quantity (Q*).
This is awesome for several reasons. First because the quantity demanded by the consumers will be exactly the same as the quantity supplied by the producers.  Which means that at that price level every consumer who wants to purchase at that price level gets to do so and every items created by the suppliers at that price level gets sold. This is good because it means we aren't wasting resources and we don't have unhappy customers. But for the businesses this is extra good because it is also where profit is maximized. 
"Okay, Mr. B," you say, "but what does any of that have to do with minimum wage?"
Hold your horses, I'm getting there.
Let me try to add one more concept, without drawing a lot of graphs and writing a lot of paragraphs. So to save time, put your finger on the graph above and pretend it is the price line. Now move your finger up or down on the graph, but be sure to keep it horizontal.
You should be able to see that at every other price, the price line will hit the supply and demand line at different locations. And thus at each of these locations either the resulting Quantity Supplied or Quantity Demanded will be greater than the other. This will result in either a surplus of a product if the price is above equilibrium or a shortage of a product if the price is below equilibrium .
"Um, Mr. B, minimum wage?"
Right, well it should come as no surprise that minimum wage works just the same way. Only because this is the labor market and not the product market we typically think of, the roles are switched around. Thus in the labor market the demand is not customers wanting to buy a product, but instead is businesses demanding employees to hire. And the supply is not businesses providing products for sale, but instead is people wanting to supply their work to the labor market. And finally the price line doesn't determine the price of a product but instead determines the wage that market should charge.
Of course there is going to be a different graph for every different market or job. So the equilibrium wage for a doctor is different than that of a plumber or a butcher or someone working the drive through at a burger joint. But if the market is left alone, the market will automatically find the equilibrium price or wage  and the market will be efficient and profit will be maximized 
This means that if the equilibrium wage for a doctor is $150,000 a year then that's what they will get paid and if the equilibrium wage for a butcher is $45,000/yr that is what they will get paid and if the equilibrium wage for a person working the drive through at a burger joint is $12,000/yr that is what they will get paid.
But then someone in the government says, "Hold on a second. You can't live on $12,000/yr. Let's make it illegal to sell your labor for less than $14,500/yr.
When someone, in this case the government, says you can't sell something below a certain price, this is called a price floor. And price floors  always cause a surplus.
"Woohoo!" you say, "That means there are more jobs than people looking for them! Right?"
Not so fast. Remember in the labor market the roles are switched and a surplus means the quantity supplied of people wanting a job will be larger than the number of employees demanded by businesses. So, what it means is that a lot of people are going to not be able to find the job they want.
Here's the graph you know you are wanting:
Pf is the Price Floor. Qd is the quantity demanded and Qs is the quantity supplied. The distance between Qd and Qs is the amount of the surplus in the market. In other words it is the difference between the number of people who want to work at a given price for a given job and the number of jobs businesses are willing to fill at that same price.
But really that number is less important than the difference between Q* and Qd. Because that number is the number of jobs that will disappear because of the price floor or in this case because of the minimum wage.
"So, then if we raise the minimum wage to $10.10/hr how many people will lose their job?" you ask.
Good question. The answer is that nobody knows for sure. Looking around you can find a veritable mixed bag of answers from some people saying that nobody will lose their jobs to others estimating that around 1 million jobs will be lost.
I obviously don't know, but I do know that I have read articles about things like McDonald's in anticipation of a higher minimum wage creating test stores in which all of the food production is completely automated and thus cutting the number of employees that they would need to hire. I've seen stories about parks that fired all of their gate attendants and replaced them with automated ticket taking machines. And if you are old enough to remember when gas stations had someone who would fill up your gas and clean your windows for free and always wondered what happen to those jobs and who's to blame for them disappearing, you need look no further than minimum wage. So, I have to think there will be some impact.
Thus, in my mind strike one against the minimum wage is that it causes jobs to go away.
"Okay, we don't know how many, but do we know what kind of jobs will go away?" you ask.
While we don't for sure what jobs will go and which ones won't, we can make some pretty good guesses. Just looking at the examples I gave above you can see that they are low skill jobs that offer low pay. They are jobs that aren't intended to be careers. They are jobs in short that are pretty much meant to be for high school teens looking for extra cash or for a place to get experience, a foot in the door as it were.
So mostly the jobs that would disappear would mean more teenagers without jobs.
"But I know of several people who are working two minimum wage jobs just to try and stay alive," you say.
I agree that's a problem, but remember that we are currently coming out of the Great Recession and that the job market is not by any means back to where it should be. When the job market and the economy is working correctly the percentage of people making minimum wage is only around 3-5% of the workforce and the vast majority of those people are, as we've already said, teenagers.
That's not surprising since as we said minimum wage is not meant to be a wage level that people earn for their career. It's an entry level position or a summer job for a teenager.
"So, if minimum wage goes up only teenagers will lose their jobs?"
Or we could say that if we got rid of minimum wage it would mostly be teenagers who saw their wages go down.
"Get rid of minimum wage? If we did that business would just offer a tiny amount for their jobs."
Nope they wouldn't. Because it works both ways. If the business offers less than equilibrium they will be giving a price below equilibrium and as we already said that will just cause a shortage. In this case we would have a large quantity of businesses demanding employees at this low wage and a very small quantity of people willing to supply their labor at such a low price.
If a burger business really did say that they were only going to offer $3.00/hr for the job of working at their drive through they would quickly find the only people willing to work there would be people who couldn't get jobs anywhere else because they were such horrible employees nobody offer a better wage was willing to hire them and the business would find their customers did really enjoy ordering from them and they would start to take their business elsewhere. In other words, the more a money a business offers the more people willing to work there and the better the pool of potential employees they have to choose from. Which means happier customers and more profits.
And I know this is true because in a normal economy only 3-5% of people are even making minimum wage. Thus most business already offer more than minimum wage. In part because their job may be more difficult, but also because they want to attract the best employee that they can afford.
Of course, I'm not saying that some jobs wouldn't lower their wages, but most wouldn't and I know most teenagers don't want to admit it, but I suspect if you asked and they were honest, most of them would admit that they could get by even if their job only paid $6.00/hr.
"What about the people who have minimum wage jobs and aren't teenagers? Lowering their pay rate might make it where they can't survive."
True, but to me the solution there is that they need to get a better job and before you get irate and start shouting about people stuck in dead end jobs and lack of skills and no money for college, etc, etc, etc. Let me say that for those people the solution to me would be to help them get skills, training and school so that they can get out of those jobs and not to artificially inflate wages across the board.
So strike two for minimum wage in my mind is that it really isn't helping the people we want it to help at all.
Finally, there's inflation.
I don't have the room to give a full rundown on inflation, but the gist is that over time the purchasing power of money can get eroded. Thus you may find that if you took $100 to the store today and bought a small pile of goods with it and then next year you went back to the same store and tried to buy the exact same pile of goods, you would find that you couldn't. Inflation would have made it to where the cost of those goods had risen. 
There are a lot of things that can cause inflation but one of the major contributors is too much money in the market place.
For instance let's pretend that all of the money in the country was only $1,000. And that in the market place the equilibrium price for bread was $1 per loaf. If we suddenly increased the amount of money in the marketplace to $2,000 what would happen?
Well since the price of bread was determined by the equilibrium price, if suddenly everyone had more money  we would suddenly see demand for products rise. In fact an increase in income is one of the things that determines that demand line we talked about way back at the first graph and as I just said an increase in income will cause an increase in demand. So what happens when you increase demand? Here's a graph that shows it:
Demand has shifted to the right (from D1 to D2) and prices have risen (from P1 to P2).
Thus giving everyone more money just means that they will drive the price of bread up. So, in our example, doubling the money supply from $1,000 to $2,000 will just cause the price of bread to double from $1 to $2.
Sure there will be a brief period in which people will have extra money in their pockets, but quickly they will find that prices have adjusted so that however much they could buy with their paychecks before the money supply doubled, that is exactly how much they would be able to buy after the money supply doubled.
This is no less true for minimum wage. But instead of doubling the amount of money in the money supply we are trying to artificially increase incomes with a different method, but the result will be the same. More money = more demand = higher prices.
To add an additional problem from the other side of the coin, if we tell the burger joint that they have to increase their employees wages by x%, the burger joint will have to raise its prices to accomplish this and so will everyone other job that hires at minimum wage and this will have a ripple effect through society. So the prices of products will also rise because of this. 
Thus in my mind the third strike against minimum wage is that in the long run we aren't really even helping the few people who do get that pay raise, because in the end the increase in demand combined with the decrease in supply will mean that inflation will raise the prices for everyone.
Again, I'm not saying that there aren't people out there working minimum wage jobs that are struggling to survive and who desperately need to get paid more. What I am saying is that those people need new jobs, not an artificial increase in the wage of their old job. They don't need to stay in a low skill, low pay, minimum wage entry level job. They need to find a better career job that they can do and if there isn't such a job then they need to improve their skills set and situation so that those jobs are accessible. And if their situation is so bad that they can't afford to do that or they are unable to do that, then society, the government, or others need to act to help them so that they can.
In summary, in my mind, raising minimum wage so that:
a) jobs disappear
b) few of the people who really need it are actually helped by it; and
c) it causes prices across the board to rise and ultimately negate any minimum wage increase anyway
seems like a giant failure.
Minimum Wage: D+
 - Also, fair warning, this obviously involves a good bit of economics and is a pretty involved topic that I'm trying to handle in a short space and as such that means I'm not going to be able to go into as much depth as I'd like in some areas. However, if you want me to explain anything further, I'd be happy to.
 - In regular English that means if I found out the total amount of CD's everyone in the market would be willing to buy at $5, $10, $15, $20 and every other possible price and I put all that in a graph, I would get the market demand for CD's. If I did the same thing at every price for people willing to sell/make CD's I would get market supply for CD's.
 - In the interest of time and expediency I'm not going to go into depth on why Supply and Demand are drawn like that. Just trust me on this one.
 - Quantity demanded is determined by where a particular price moving horizontally across the graph strike the Demand line. Quantity supplied is determined by where a particular price moving horizontally across the graph strikes the Supply line. At equilibrium those are the same place.
 - And anyone who tells you that a business is in business for any reason other than making money and maximizing profits is probably selling you on something.
 - If William Shatner shows up you're thinking of the wrong kind of price line.
 - A surplus is bad because it means we've wasted resources and have extra products sitting around being unbought and a shortage is bad because it means that some customers are not getting to buy when they wanted to. But even more importantly the seller is not maximizing profit.
 - Invisible hand ftw
 - Profit is maximized in the market, not necessarily in any one person's pocket.
 - Which translates to roughly $7.25/hr and obviously minimum wage didn't start at $7.25/hr and if you want to find who really started minimum wage in the US, you can pretty much point that finger at FDR.
 - That are set above equilibrium price
 - Historically the inflation rate in the US tends to be around 2-3% per year.
 - Or even if a significant portion of the people had more money
 - This is called the Wage-Price spiral and to see the beginning of its effects without me drawing another graph, this is the opposite of the increase in demand. It is a decrease in supply. To see the effect put your finger on the graph over the supply line and then move it to the left. Note how the spot where the demand line and your finger cross moves up. An increase in prices.
 - Minimum wage's heart is in the right place. So I didn't give it an "F".