Metaphysics is a notoriously difficult term to define, but for purposes of briefly introducing it to nonphilosophers, it can be identified as the study of any of the most fundamental concepts and beliefs, on which many other concepts and beliefs rest--concepts such as being, existence, universal, property, relation, causation, space, time, event, and many others.
The field of metaphysics has changed a lot since it first received its name by Aristotle's editors (see below), over the many centuries of its existence. Problems that were not originally considered metaphysical were added to metaphysics. Other problems that were for centuries considered metaphysical problems are now typically relegated to their own separate subheadings in philosophy. For example, there are a lot of problems that used to be considered part of metaphysics, but are now more commonly considered parts of the philosophy of religion, philosophy of mind, philosophy of perception, and philosophy of science. It would require quite a long time to state all the problems that have, at one time or another, been considered part of metaphysics.
What might be called the core metaphysical problems would be the ones which have always been considered metaphysical and which have never been considered not metaphysical. What most of such problems have in common is that they are the problems of ontology, "the science of being qua being" (see the ontology article for an explanation of what this means).
The origin of the word 'metaphysics'
The ancient Greek philosopher Aristotle wrote a number of books which together were called the Physics. An early edition of the works of Aristotle were organized in such a way that there was another set of books that were placed right after the Physics. These books seemed to concern a basic, fundamental area of philosophical inquiry, which at the time did not have a name. So early Aristotle scholars called those books "ta meta ta physika," which means "the (books that come) after the (books about) physics." That, then, is the origin of the word 'metaphysics'.
Hence, etymologically speaking, metaphysics is the subject of those books by Aristotle which were called, collectively, the Metaphysics. So, etymologically, what 'metaphysics' means is 'the subject that the Metaphysics of Aristotle is about'.
What were those books by Aristotle about? The Metaphysics was divided into three parts, called (1) ontology, (2) theology, and (3) universal science. So ontology, theology, and universal science are regarded as the three traditional branches of metaphysics. (1) 'Ontology' is the study of existence; it has been traditionally defined as 'the science of being qua being'. (2) 'Theology' means, here, the study of God or the gods and of questions about the divine. (3) 'Universal science' is supposed to be the study of so-called first principles, which underlie all other inquiries; an example of such a principle is the law of non-contradiction: "A thing cannot both be and not be at the same time, and in the same respect." A particular apple cannot both exist and not exist at the same time. It can't be all red and all green at the same time. So that was the Aristotelian conception of metaphysics.
Metaphysical subdisciplines (i.e., fields now or traditionally treated as part of metaphysics):
Metaphysical topics and problems