A traditional economic system is the best place to start because it is, quite literally, the most traditional and ancient type of economy in the world. There are certain elements of a traditional economy than those in more advanced economies, such as Mixed, would like to see a return to prominence.
In terms of economic advancement, the command economic system is the next step up from a traditional economy. This by no means indicates that it is fairer or an exact improvement; there are many things fundamentally wrong with a command economy
A market economy is very similar to a free market. The government does not control vital resources, valuable goods or any other major segment of the economy. In this way, organizations run by the people determine how the economy runs, how supply is generated, what demands are necessary, etc
A mixed economic system (also known as a Dual Economy) is just like it sounds (a combination of economic systems), but it primarily refers to a mixture of a market and command economy (for obvious reasons, a traditional economy does not typically mix well). As you can imagine, many variations exist, with some mixed economies being primarily free markets and others being strongly controlled by the government. Learn more about an essential part of our economy with this free post on understanding the stock market.