We can never be certain that the processing function will call the callback with the correct arguments. Callbacks have two fundamental flaws: Firstly, they are not type-safe. The processing function then calls the callback when appropriate. A callback is a pointer to a function, so if you want a processing function to notify you about some event you pass a pointer to another function (the callback) to the processing function. Older toolkits achieve this kind of communication using callbacks. For example, if a user clicks a Close button, we probably want the window's close() function to be called.
More generally, we want objects of any kind to be able to communicate with one another. In GUI programming, when we change one widget, we often want another widget to be notified. The signals and slots mechanism is a central feature of Qt and probably the part that differs most from the features provided by other frameworks.
Signals and slots are used for communication between objects.