An unincorporated one-person business is called a sole proprietorship.
The person who owns a sole proprietorship is called a Sole Proprietor and is considered self-employed (see Self-Employed Individuals), rather than an employee. (See Employee vs. Independent Contractor.)
A sole proprietorship is the simplest form of business entity because you have no separate organizing document such as articles of incorporation. But your Record Keeping responsibilities for tax purposes are the same as with other business formats.
The main downside to the sole proprietorship is that the owner has unlimited liability if the business is sued. That's why some one-person businesses organize as a corporation or a single-member Limited Liability Company (LLC).
Sole proprietors report their business income and expenses on Schedule C of IRS Form 1040. Also, sole proprietors pay Self-Employment Tax in lieu of Employment Taxes.