| In the United States, a "foreign corporation" is a corporation operating in a state other than the state where it is legally incorporated. A corporation based in New York, for instance, would be a foreign corporation under New Jersey law if it started operations in New Jersey. If it re-incorporated in New Jersey, but continued its business in New York, it would become a foreign corporation under New York law.
Usually, foreign corporations have to register with a state's government before they can do business in that state. The commerce clause of the Constitution implicitly prohibits states from regulating corporations that are using that state for commerce between two other states (New Jersey cannot force a New York corporation to register just because it uses New Jersey airspace to fly goods from New York to California). However, if a corporation has minimum contacts within a state on a regular basis, and those contacts justify subjecting the corporation to state law, then the corporation can be made to register.
The particulars of registration vary from state to state. Usually, the corporation has to fill out an application and provide a filing fee. More importantly, the corporation has to designate a registered agent to receive any claims filed against the corporation in state court.
Many Fortune 500 companies incorporate in Delaware (for various legal reasons), and then do almost all of their business in other states. Thus, they exist almost entirely as foreign corporations. | 
