Here are the definitions:
Firm
A partnership or association for carrying on a business. This is tyypically used to describe a group of lawyers in business together.
The law firm of Branston and Jones.
Business:
This is much more general, and can be used as an adjective (She enrolled in a good business school.)
1. an occupation, profession, or trade: His business is poultry farming.
2. the purchase and sale of goods in an attempt to make a profit.
3. a person, partnership, or corporation engaged in commerce, manufacturing, or a service; profit-seeking enterprise or concern.
4. volume of trade; patronage: Most of the store's business comes from local families.
5. a building or site where commercial work is carried on, as a factory, store, or office; place of work: His business is on the corner of Broadway and Elm Street.
6. that with which a person is principally and seriously concerned: Words are a writer's business.
7. something with which a person is rightfully concerned: What they are doing is none of my business.
Enterprise:
Also can be formed as an adjective: He is an enterprising young man. (He is a serious businessman). It focuses on the character of energy and risk-taking.
1. a project undertaken or to be undertaken, esp. one that is important or difficult or that requires boldness or energy: To keep the peace is a difficult enterprise.
2. a plan for such a project.
3. participation or engagement in such projects: Our country was formed by the enterprise of resolute men and women.
4. boldness or readiness in undertaking; adventurous spirit; ingenuity.
5. a company organized for commercial purposes; business firm.
Company:
1. a number of individuals assembled or associated together; group of people.
2. a guest or guests: We're having company for dinner.
3. an assemblage of persons for social purposes.
4. companionship; fellowship; association: I always enjoy her company.
5. one's usual companions: I don't like the company he keeps.
6. society collectively.
7. a number of persons united or incorporated for joint action, esp. for business: a publishing company; a dance company.
Corporation:
I would put this in the list of terms related to businesses. It is legal and formal.
A body that is granted a charter recognizing it as a separate legal entity having its own rights, privileges, and liabilities distinct from those of its members.
The most important aspect of a corporation is limited liability. That is, shareholders have the right to participate in the profits, through dividends and/or the appreciation of stock, but are not held personally liable for the company's debts.