The most basic business form is a sole proprietorship, which is typical for small businesses that don't have a need to distinguish between the company and the business owner. A partnership is similar to a sole proprietorship, except it includes two or more partners.
There's also the limited liability company (LLC), which gives small business owners protection from a potential lawsuit and separates the finances of the owner and the business.
Finally, there are two kinds of corporations: S corporations and C corporations. They are more complicated business structures. Corporations have distinct rules about shareholders and taxation that make them more common for larger private businesses than small ones.
It's also worth understanding the different stages in private company life cycles. The earliest stage for a private company is a start-up, which could refer to a small company or one that is funded by venture capital and aims to go public in a few years. By contrast, there are private companies that have been that way for decades; they may be family-owned businesses and are likely to stay private.