Final answer:
The agreement between the broker and the home seller is known as an implied contract or a verbal agreement, formed by actions suggesting an agreement.
Step-by-step explanation:
When a broker brings a buyer to a home that has been advertised as for sale by owner, and discusses sales-related matters and offers advice that the owner accepts, all without anything being written down, the type of agreement that exists is known as an implied contract or a verbal agreement.
An implied contract is formed by the actions of the parties that suggest they have reached an agreement, despite not having a written contract. However, in real estate transactions, agreements should typically be in writing to be enforceable due to statutes requiring real estate contracts to be in writing, such as the Statute of Frauds.