A company is a form of business organization.
In the United States, a company is a "corporation—or, less commonly, an association, partnership, or union—that carries on an industrial enterprise." Generally, a company may be a "corporation, partnership, association, joint-stock company, trust, fund, or organized group of persons, whether incorporated or not, and (in an official capacity) any receiver, trustee in bankruptcy, or similar official, or liquidating agent, for any of the foregoing."
So, a corporation is a form of company whose articles of incorporation have been approved in some state
In business we typically structure companies under state law in one of these forms:
Corporations (have shares and shareholders)
Limited Liability Companies (LLCs) (have shareholders or members)
Limited Partnerships (LPs) (have General Partners and Limited Partners)
Limited Liability Partnerships (LLPs) (have general and limited partners)
Sub Chapter S Corporations have for all intensive purposes been replaced by LLCs
It is incorrect to say that a company has unlimited liabilities. It depends on how it is structured. It is however correct that the corporation can shield its shareholders from liabilities.