When two or more parties share the profits and liabilities of a company it is called a partnership.
Partnerships usually are "limited partnerships" in which a managing partner runs the day-to-day business while the limited partners are not involved in management decisions and enjoy limited liability.
Sole proprietorships and partnerships usually adopt a company name known as a DBA or "doing business as" that company name. As business grows, the principals might want to apply for a corporate charter.
The corporation formed in this process is considered to be a separate individual by law and usually provides certain tax relief, expense deductions, and liability protection to the principals involved. Corporations may be privately or publicly owned.