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.
Market economy is one of the two pure models of economic organization, the other one being command economy, or centrally planned economy. In market economy the the business firms are run and managed as independent business organizations that compete with each other. Questions like what goods and services these businesses produce, how they produce it, what prices they charge, and to whom they sell these products is determined by the market forces of demand and supply. The business exist in such an economy primarily for the purpose of making profit for themselves.
In contrast, in command economy the manufacturing and supply activities are controlled and managed by a central authority like a government on behalf of the people. Business decisions are taken by this central authority to maximize the good of the people. There is no competition between different business firms. The decision of businesses are taken on their behalf by a central planning system that aims to maximize the total good of the society rather than the profit of the businesses. Though, separate businesses seem to exist in the command economy, these are all like different units of the same entity rather than different entities competing with each other. Business firms still buy and sell. But the decision on what is produced and sold. to whom it is sold, and the prices is dependent on the central planning system rather than the interplay of market demand and supply forces.
It is worthwhile noting that both market and command economies are theoretical models that do not exist in absolute pure form. All the countries have economies that are a mix of the two types. Economies that are generally described as market economy are, in reality, mixed economies in which the features of market economy are more dominant as compared to those of command economy.