What's the relation between business and a market economy?

I think that
the answer you are looking for is that a market economy allows people to form their own
businesses to sell pretty much whatever they want (as long as they do not try to sell something
illegal).  In a market economy, choices like whether to create a business or what to have your
business make and sell are left up to the individual.

This is why a market
economy is often referred to as a free market economy.  It is an economy where people are free
to make economic decisions based on what they feel is best.  This is different from, for
example, a command economy where a government would be deciding what businesses could exist,
what they could sell, what their prices would be, etc.

So businesses are
freest to exist and to make their own decisions in a market economy.