Economists generally recognize three distinct types of economic system. These are 1) command economies; 2) market economies and 3) traditional economies. Each of these kinds of economies answers the three basic economic questions (What to produce, how to produce it, for whom to produce it) in different ways.
In a command economy, the government decides the answers to the three basic questions. It decides what will be made, how they will be made, and who will get them. Recently, pure command economies have usually been communist countries. Good examples today would be North Korea and China.
In a market economy, consumers decide the answers to the three questions. They do this by their choices of what to buy. No one tells companies what to make -- they make whatever they think will sell. If they choose wrong, they go out of business. Most developed economies today are predominantly market economies. The US, Japan and Germany are all market economies.
In a traditional economy, the three questions get answered by referring to tradition -- you make what has always been made, in the way it has always been made, etc. There aren't really any countries whose whole economies are traditional. The closest you could get to this would be Afghanistan or Bhutan -- places where there is little connection to the global economy.