The economy can be thought of as a system of production and exchange of goods and services among people and organizations (private or public). The money, price and financial system are there to facilitate production and exchange, and provide a common unit of value to different goods and services.
Traditionally, economics can be thought of as either the study of the economy or the study of rational human choice. There is actually no set answer.
Ha-Joon Chang, an economist at the University of Cambridge, provides a good explanation in the first few pages of his book Economics: The Users Guide:
In [one] view, economics is defined by its theoretical approach, rather than its subject matter. Economics is a study of rational choice, that is, choice made on the basis of deliberate, systematic calculation of the maximum extent to which the ends can be met by using the inevitably scarce means. The subject matter can be anything – marriage, having children, crime or drug addiction…
[Another view] is that economics should be defined not in terms of its methodology, or theoretical approach, but in terms of its subject matter…The subject matter of economics should be the economy – which involves money, work, technology, international trade, taxes and other things that have to do with the ways in which we produce goods and services, distribute the incomes generated in the process and consume the things thus produced