What are the advantages and disadvantages of a market economy?

A true market economy exists when supply and demand control the amount of goods and services in the market. There is no government interference in a true market economy. This has never happened before, as governments have always at least used tariffs and internal taxes in order to regulate commerce; however, there have been times that government has been less active in managing the economy than it is today.
This economy has several advantages. People are free to produce and buy as many goods as their income will allow. A person is free to decide what to do with his/her money.
This economy also has many disadvantages. There are no protections for consumers. They are not protected against business fraud or faulty products. There are no protections for workers either, as they are only paid what the market dictates. While they are free to take other jobs, they may be limited in their options due to ability or location. The market can also produce periods of economic success followed by extreme market downturns.
The unregulated market has never existed in the United States, but it closely resembles the period of the Gilded Age in which a minority of Americans controlled most of the nation's wealth. While there were tariffs to control the prices of imports, businesses could pay workers anything, and businesses could also create products regardless of safety. It was a great time for large business but a horrible time to be in the working class or a consumer.

A market economy is one in which the prices of goods, investments, and services are unimpeded by government regulation of any sort with the market and laws of supply and demand setting prices.
The first thing to note is that a pure market economy has never existed. All governments engage in various practices such as taxation, infrastructure building, subsidies, and other activities which affect pricing. For example, if a government pays for roads via taxation but not for railroad tracks, the government is in effect creating an artificially low price for road vs. rail transport.
The good point of a market economy is that it should, in theory, assuming a perfectly competitive market, allow for prices to rise and fall with consumer demand. 
Unfortunately, the assumption of a perfect market ignores such issues as monopoly or collusion, i.e. businesses setting prices artificially low to drive competitors out of business and then forming monopolies or oligopolies which keep prices artificially high. Also, a free market does not account for ethics or human costs. In a perfectly free market, for example, businesses could pay children a few dollars a day to work 16 hour shifts in unsafe conditions.