There are 4 main types of business organization: sole proprietorship, partnership, corporation, and Limited Liability Company, or LLC.
A business is defined as an organization or enterprising entity engaged in commercial, industrial, or professional activities. Businesses can be for-profit entities or they can be non-profit organizations that operate to fulfill a charitable mission or further a social cause.
The term is also often used colloquially (but not by lawyers or by public officials) to refer to a company. A company, on the other hand, is a separate legal entity and provides for limited liability, as well as corporate tax rates. A company structure is more complicated and expensive to set up, but offers more protection and benefits for the owner.
Cooperative: Often referred to as a "co-op", a cooperative is a limited-liability business that can organize as for-profit or not-for-profit. A cooperative differs from a corporation in that it has members, not shareholders, and they share decision-making authority. Cooperatives are typically classified as either consumer cooperatives or worker cooperatives. Cooperatives are fundamental to the ideology of economic democracy.
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Companies formed by letters patent: Most corporations by letters patent are corporations sole and not companies as the term is commonly understood today.
Charter corporations: Before the passing of modern companies legislation, these were the only types of companies. Now they are relatively rare, except for very old companies that still survive (of which there are still many, particularly many British banks), or modern societies that fulfill a quasi-regulatory function (for example, the Bank of England is a corporation formed by a modern charter).