In Common Lisp there are several operators that are referred to as “special”.  A special operator is one which has different behaviour in terms of bindings or flow control.  The list can be found, for instance, here.  The programmer cannot create their own special operators, there is no way to create a new entity for which special-operator-p returns non-nil.

Some examples will, perhaps, clarify what this means.  The Common Lisp if special operator takes two or three arguments.  The first is the condition, the second is evaluated only if the condition evaluates to non-nil, and the optional third argument is evaluated only if the condition evaluates to nil.  Hidden within the if form, there is a flow control that decides whether or not to evaluate certain arguments.

In the context of bindings, catch sets up a dynamic binding that becomes important if code below it invokes throw.  The labels special operator sets up a lexical binding within its scope.

The special-operator-p function returns non-nil if the symbol passed as its argument corresponds to a special operator.  Now, when might the programmer need to use this?  Most obviously in a code walker, something that reads and analyses Lisp code.  I have never seen it used other than in informational contexts, to label a particular piece of code as containing a special operator, or by the implementation to forbid the redefinition of such operators.  I haven’t seen a context in which complex actions are initiated by special-operator-p returning non-nil, only messages, errors, or skipping over a block while analysing code.