In Australia, it is generally only possible to stop another person trading under a particular ‘brand’ name by relying on trade mark rights.
Trade mark rights can be registered and/or unregistered.
A registered trade mark is granted by IP Australia. It gives a person the right to use that trade mark Australia wide (subject to some limitations), and the right to stop others user the same or similar trade mark.
A registered trade mark is infringed when another person uses an identical or deceptively similar mark, for the same goods or services as covered by the trade mark registration (or similar goods or services).
Unregistered trade mark rights are obtained through use of a particular trade mark. However, they are more difficult to enforce. This is because the person claiming to have unregistered trade mark rights must usually prove that they have a reputation in their trade mark for particular goods and/or services.
An unregistered trade mark can be enforced in two ways, namely by taking action under the Australian Consumer Law (for misleading or deceptive conduct) and/or by using the tort of passing off.
Both of these actions are very similar.
The law of passing off prevents one person from misrepresenting that their goods or services are those or another, or have some association with another trader. The law exists to protect the goodwill that has been built up in a particular business.
The Australian Consumer Law also prohibits such activities, though there are subtle differences (the main one being that the Australian Consumer Law is designed to protect consumers, and not goodwill in businesses).
When someone is found liable for having infringed trade mark rights, they can be ordered to pay damages, or an account of their profits. They can also be ordered to stop using the infringing trade mark.