Who answers the three economic questions in a market economy? How is this different from command economy?

Market economies are characterized by their free markets. Examples are the United States and the United Kingdom.

That's pretty clear. What it means is that they trust in the free market and its operation to answer the questions of what to produce, how much to produce, and for whom.

Command economies are characterized by their control of markets. Examples are China and North Korea.

They rely on the government, not the free market itself, to control what should be produced. This makes for a more strict system that can pay off in the long run, but can also run the risk of being heavy-handed.