A market economy is based on the principles of supply and demand, and lets business survive or fail without much interaction from the government. A pure market economy is impractical to implement, most market economies around the world have a component of government influence.
The theory behind a market economy is that the market will respond to changes in demand with changes in supply. Through competition, the price of products will reach a suitable equilibrium. In mixed economies like the United States, the government influences supply and demand by providing subsidiaries, controlling the prices of certain goods and regulations in certain industries.