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.