An Australian company is its own legal entity and will end in Pty Ltd, or Ltd. A company is required to have at least one officeholder (director), who is responsible for managing the company.
Different company types have different director requirements, for example, a public company must have at least 3 directors, section 201A has the requirements.
A business name is the name your business operates under. You need to register a business name if you conduct business under a name other than your own. You need to register a business name if you conduct business under a name other than your own or wish to use your company name without the legal elements from your company name (e.g. PTY LTD).
For more information, see What is a business name?