What is the difference between free-market and centrally planned economies?
First, we should note that there really aren't totally free market economies anymore than there are economies where all activities are centrally planned. So these are basically ideal types, used to illustrate the logical conclusions of two very different philosophical approaches to the proper relationship between the state and economic activity. A "planned" economy is one in which the state, or government, makes the basic economic decisions. Usually this is achieved through direct government ownership of factories, farms, and other means of protection. Even transactions between individuals, or individuals and businesses, can be regulated by price controls, rationing, or other means. Often production quotas determine how much of a particular good is produced, and the state might, through education systems, determine what jobs an individual might pursue. The point is that the most important economic decisions are made by the state.
In a market economy, the basic economic decisions are made by individuals, or by larger companies or businesses. Business owners determine how much they want to produce, how much they are willing to pay people to work, and so on. Individuals determine how much they are willing to pay for items, to whom they will sell their labor (in other words, who they will work for), and how they want to spend the money from their labor. We often say that people are guided in a market economy by the profit motive and by the larger forces of supply and demand. Advocates of free market economic principles argue that these forces regulate the economy in a way that maximizes economic efficiency and benefits the most people. If people don't want to work for low wages, they can take their labor to another employer. If they don't want to buy an item for a certain price, they can refuse to do so, and if enough people make the same choice, then businesses will have to sell that item at a price people are willing to pay. Again, the basic decisions are made by individuals making rational decisions, the sum total of which we call the "market."
Market economies and centrally-planned economies, often known as command economies, are very different. In a market economy, the activity is unplanned. Meaning people determine what occurs in the economy. What to produce, what to sell, where to produce, how much to produce, and how much to sell a product for all determined by the market, or the supply and the demand. In a market economy, the means of production are privately owned by individuals or companies, and there are voluntary exchanges and contracts. This means that businesses own the resources and are free to create their own exchanges and contracts without government decrees. Prices are determined naturally through supply and demand. In a command economy, all of the means of production (land, capital, and resources) are owned by the government. Government officials determine when, where, and how much to produce at one time. Prices do not arise naturally through supply and demand, as the government determines the supply and sets the demand. It is also important to note that profits in market economies determine resource allocation, where in a command economy political factors determine resource allocation.