Domestic and foreign in this context refers to the State where the LLC is created. A company which is registered as an LLC in Michigan and which does business in Michigan is operating as a domestic LLC. If the same company is doing business in Illinois (a physical presence), it is operating as a foreign LLC in the other State.
This commonly comes up when LLCs are formed in States with business-friendly tax laws but do business in their home State. It may also occur when a business starts expanding into other States. A foreign LLC is required to register with the Secretary of State in the foreign State. The LLC certificate (and other documents) issued by the home State will be needed.
The purpose of registering is to meet regulatory and tax requirements of the foreign State.
Operating in a State generally means:
Having a bank account in that State. Selling in that State through some party directly tied to the LLC (a distributor or rep). Owning property (real estate or a fleet of trucks or example) in the State. Having offices, facilities or holding regular meetings in the State.
These hallmarks also vary by State and should be available at the Secretary of State’s office.