A subsidiary is a company whose more than 50% of voting stock is controlled by another company known as the parent company.
A foreign subsidiary company is a partially or even a wholly owned company which is a part of another large corporation. This large corporation must have its headquarters in another country. The foreign subsidiary company work as per the laws the country in which they are located.
The parent company however has to carry the financials of the foreign subsidiary on its books. The foreign subsidiary companies have to obey the laws of the country they are operating in.