When you start a business for profit (whether you a make profit or not) there are principal methods under which you can trade. These are:
• As a sole trader
• In a partnership
• A limited Liability Partnership (LLP)
• As a limited liability company (LTD)
• As a Corporation (inc) (USA)
• Cooperative (USA)
• S Corporation (USA)
Not for profit and Charities have other methods which are beyond the scope of this guide.
The rules for trading will vary from country to country and you need to check what is permitted in your country.