The market economy is an economy type where the private businesses dominate. The economy is open, the trade is on a large scale, and very often big percentage of the produced goods are exported, while others are imported.
The command economy is in the hands of the government, which controls everything in it, and has all the resources in its hands. The economy is usually more closed, and there's trade with only a handful of countries, usually with the same economic type. The private businesses do not exist.
By combining this too we get mixed economy. In this type of economy, the government still has big portion of the resources in its hands, and is still heavily involved in the decisions, but the economy is much more open for trade. The amount of production is much bigger than what the country needs, so there's lot of goods that are exported. The benefit is that the government has control over the most important things in the country, and the workers are protected, as well as making big profit from the export of goods while maintaining the balance on home ground. A perfect example of this is China.
