What is political economy?

Political economy was, in some ways, an outgrowth of the transition from monarchy to a parliamentary system of government, and more particularly, government finance.
In antiquity, the science of organizing cities was "politics," and included systems of law to regulate behavior (e.g. rules concerning murder and theft) and external relations (diplomacy, military matters). There was little in the way of what we would now think of as civil service and most public services were provided by liturgies (assigning wealthy people, on a rotating basis, the responsibility for providing services or financing for public projects). Economics had to do with administering households, and had a far more practical and financial focus. 
In the Renaissance, nations were run as part of a monarch's household, and thus what we would now think of as national finance or economic planning (taxation, public works, military funding, etc.) was technically organized as a department of royal household finance. As the power of Parliament increased and that of the monarchy decreased, economics became public, pertaining to the polis or state as a whole, rather than run as part of the household economy of the monarch. The term political economy proliferated in the eighteenth and nineteenth centuries to reflect this shift.