Clients
A client is a user or system that sends or receives a notification. The client may be a physical person, or may just be a component in a computer system. Either way, a client is not a core part of FME Server, rather someone or somthing that interacts with it.
For example, a database update might cause a trigger to send a notification to FME Server, in which case the database system is the client. But a client could also be a person who, for example, triggers a notification by sending an email to FME Server.
Likewise, FME Server can send a notification for another client system to receive. Alternatively this client can also be a real person, who might receive a notification in the form of an email.
Publishers vs Subscribers
Clients that send a notification to FME Server are called Publishers.
Clients that receive a notification from FME Server are called Subscribers.