What is Foreign Corporation?

Legal Definition
A foreign corporation is a term used in the United States for an existing corporation that is registered to do business in a state or jurisdiction other than where it was originally incorporated. A foreign corporation is one incorporated as a domestic corporation in one state of the United States, authorized to do business in additional state(s); the term is also applied to a corporation incorporated outside the United States which is authorized to do business in one or more states of the United States.

To a degree, the same rules apply with respect to a Limited Liability Company (LLC), in that it is a domestic LLC in the state where it is originally chartered, and a foreign LLC everywhere else.

For U.S. federal tax purposes, "foreign corporation" means a corporation which is not created or organized in the United States.
-- Wikipedia
Legal Definition
A corporation that does business in a state but is incorporated in a different state or a foreign country. A foreign corporations must file a notice of doing business in any state in which it does substantial business.

Compare to a domestic corporation, which is incorporated in the state it is doing business in.
See also
Legal Definition
A firm that incorported under foreign law and is owned by that nonlocal government. AKA alien corporation, foreign concern, and foreign firm.
Legal Definition
A corporation formed under the laws of another state or country and unincorporated companies of foreign countries having corporate attributes. See 77 U. S. 566, 19 L. Ed. 1029.
-- Ballentine's Law Dictionary