What is the purpose of setting a minimum wage?

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What is the purpose of setting a minimum wage?

(This is an offshoot of the "How can American IT Professionals Compete with asian workers?" discussion)

And how is it determined?

For what area is it determined? Some places cost more to live than others (sometimes even within the same city), so what do you do? Set it to be minimum for the 'cheapest to live in' place and let those in more expensive places suffer? or set it for the most expensive place, and give those who live in cheaper places a windfall? or do you tell people they can't live in certain places where the cost is too high?

And who do you set it for? Do you set it to suit a single young man? a married couple, no children, both work? a maried couple, two children, one works? a single mother who has to pay a baby sitter? (and does the babysitter get minimum wage too?) or do you have different minimum wages for different people in different circumstances?

And how are businesses supposed to pay for it (especially small businesses)? Does he raise his prices? Lay off some workers? And if he raises prices, doesn't that defeat the purpose? Soon prices on everything will be higher and the new minimum wage will have the same purchasing power as the old one does now, wouldn't it?

And finally is there anyone else who has thought all of these things through?