When designing software (think UML diagrams for example) and real world objects.
How does one identify a suitable case for an Abstract class?
For example if we had an [Employee] and [Fireman] and [paidFireman] and [unpaidFireman]...I am having trouble seeing whether a Fireman or Employee should be abstract and why?
There are many uses for an abstract class. An abstract class is one that cannot have any direct instances.
In software design, it is one way to describe an interface. Some of the declared operations can be implemented in the superclass. Any remaining implementations must be specified in sub-classes. Regardless of where the implementations exist, an abstract class means there can be no direct instances, only instances of some non-abstract subclass.
In a domain analysis, an abstract class is a way of modeling an abstraction. For example, think of the abstraction Role. It is useful to say that a Person plays a number of Roles. However, there is no instance of a Role that makes sense, without it also being a more specific kind of Role, such as Employee, Fireman, or Teacher. For this situation, you not only want Role to be abstract, you also want a covering axiom. For more on that, please read https://stackoverflow.com/a/35950236/2596664.
Abstract classes are one of those more esoteric constructs in UML. Since classes are already an abstraction of real world things, an abstract class is even one level higher. Abstract classes can not be instantiated (since it is assumed they miss something for a real life). Whether you say that Fireman is abstract while the paid/unpaid are not, is a pure point of view and must be argued in the specific domain.
As a rule of thumb: leave abstract classes out of the door until you come to a point where you feel the urgent need for it. Introducing abstractness limits your model (and can help to avoid some malformed results of it). But without those limits the model is still valid as long as the architect sticks to common sense rules.
It mainly depends on your functional requirements.
If it makes sense in your application just to have simple employees (without designating them as firemen, policemen, or craftsmen), then the class may not be abstract, as the application will have to make instances just of the Employee class.
If that doesn't make sense, i.e. the occupation of each of your employees needs to be known at creation time, abstract classes come into consideration. But still they aren't necessary in every case. The easiest way to make sure the occupation is known is to model it as a mandatory attribute. Introducing a subclass only makes sense if there is specialized behavior for each of those subclasses. If, e.g., the salary of the firemen is calculated as 50$ * count of the fires he exstinguished, but the salary of the policemen is 1000$ + 50 * rank, then you model an abstract operation getSalary() in the Employee class, which will be concretely specified and implemented in each of the subclasses.
As the concept of interface also got mentioned in one of the answers, an interface describes the obligation to implement certain operations in all classes realizing that interface. That's much the same as an abstract operation in an abstract class. But the abstract class can contain much more than an interface: attributes and non-abstract operations.
So the rule of thumb is: For concepts of your domain for which interface and behavior can be fully described, use non-abstract classes. For concepts for which only interfaces and no behavior can be described, use interfaces. For concepts for which interfaces and part of the behavior can be described, use abstract classes.
