The dictionary refers to a mediator as a neutral party that assists in negotiations and conflict resolution. In our world, a mediator is a behavioral design pattern that allows us to expose a unified interface through which the different parts of a system may communicate.
If it appears a system has too many direct relationships between components, it may be time to have a central point of control that components communicate through instead. The Mediator promotes loose coupling by ensuring that instead of components referring to each other explicitly, their interaction is handled through this central point. This can help us decouple systems and improve the potential for component reusability.
A real-world analogy could be a typical airport traffic control system. A tower (Mediator) handles what planes can take off and land because all communications (notifications being listened out for or broadcast) are done from the planes to the control tower, rather than from plane-to-plane. A centralized controller is key to the success of this system and that's really the role a Mediator plays in software design.
In implementation terms, the Mediator pattern is essentially a shared subject in the Observer pattern. This might assume that a direction Publish/Subscribe relationship between objects or modules in such systems is sacrificed in order to maintain a central point of contact.
It may also be considered supplemental - perhaps used for application-level notifications such as a communication between different subsystems that are themselves complex and may desire internal component decoupling through Publish/Subscribe relationships.
Another analogy would be DOM event bubbling and event delegation. If all subscriptions in a system are made against the document rather than individual nodes, the document effectively serves as a Mediator. Instead of binding to the events of the individual nodes, a higher level object is given the responsibility of notifying subscribers about interaction events.