Typically, people will argue that the US has a CAPITALIST economic system, but the US actually has a mixed economy with some aspects of capitalism, some aspects of corporatism, some aspects of socialism, and some aspects of regulatorily-managed economics.
In terms of capitalism, there are many kinds ranging from laissez-faire capitalism to social capitalism, where laissez-faire involves no government regulation of businesses, corporations, commerce, or production and social capitalism involves heavy government regulation of businesses, corporations, commerce, and production.
The US has encompassed all ends of the spectrum during its history.
As of right now, the US leans much more towards social capitalism than laissez-faire capitalism with programs like a graduated income tax, inheritance tax, social security, welfare, etc. Government branches of regulation include the FDA, FTC, FCC, Department of Commerce, Department of Agriculture, etc.
The United States operates under a capitalist (also referred to as "free enterprise") mixed economic system, where there is little government interference in private trade (unless the trade violates the Constitution or other federal laws).
The United States has a mixed economic system. This means that it consists of a command and free market economy.