If you are the only owner of the business, you are a sole proprietor. There is no government filing or approval required to operate your business as a sole proprietor. Millions of Americans do it this way. Just report the profits and losses from the business in your tax return on IRS Form 1040 on a separate sheet called Schedule C. And pay your federal and state income taxes if the business earns a profit.
If you have a business partner, the business is a partnership. That means that you and your partner share responsibility for running the business and share the profits and losses. Exactly how you want to do this should be spelled out in a partnership agreement.
Many business owners decide to operate as a limited liability company (LLC) or to incorporate their businesses. This is not a legal requirement. But it may be a good idea to form an LLC or a corporation because it helps protect you personally from debts and liabilities connected with the business. It is perfectly acceptable to form a separate legal entity like a corporation or LLC even if there is only one owner (shareholder or LLC member).