Part of the trouble with defining metaphysics lies in how much the field has changed since it first received its name by Aristotle's editors centuries ago (see below). Problems that were not originally considered metaphysical have been added to metaphysics. Other problems that were considered metaphysical problems for centuries are now typically relegated to their own separate subheadings in philosophy, such as philosophy of religion, philosophy of mind, philosophy of perception, philosophy of language, 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".
Other philosophical traditions have very different conceptions of the metaphysical problems than those in the Western philosophical tradition; for example, Taoism and indeed, much of Eastern philosophy completely reject many of the most basic tenets of Aristotelian metaphysics, principles which have by now become almost completely internalized and beyond question in Western philosophy, though a number of dissidents from Aristotelian metaphysics have emerged in the west, such as Georg Wilhelm Friedrich Hegel's Science of Logic.
The ancient Greek philosopher Aristotle produced a number of works which together were called the Physics. In an early edition, the works of Aristotle were organized in such a way that there another set of works was 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 Aristotelian scholars called those books τὰ μετὰ τὰ φυσικά, "ta meta ta physika", which means "the (books that come) after the (books about) physics." That, then, is the origin of the word 'metaphysics' (in Greek, μεταφυσικά).
Hence, etymologically speaking, metaphysics is the subject of those books by Aristotle which were called, collectively, the Metaphysics. Technically, it was so named because it came after the book of Physics. But the actual subject matter in the book, perhaps coincidentally, are on the topic of things that underly the physical -- "beyond" the physical, so to speak -- therefore fitting the word in two ways.
The Metaphysics was divided into three parts, now regarded as the traditional branches of Western metaphysics, called (1) ontology, (2) theology, and (3) universal science. There were also some smaller, perhaps tangential matters: a philosophical lexicon, an attempt to define philosophy in general, and several extracts from the Physics repeated verbatim.
Ontology is the study of existence; it has been traditionally defined as 'the science of beingqua being'.
Theology means, here, the study of God or the gods and of questions about the divine.
Universal science is supposed to be the study of so-called first principless, 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. Universal science or first philosophy treats of "being qua being"--that is, what is basis to all science before one adds the particular details of any one science. This includes matters like causality, substance, species, and elements.
The apple is an excellent example of a physical object: one can pick it up, throw it around, eat it, and so on. It occupies space and time and has a variety of properties. Suppose we ask: whatare physical objects? This might seem like the sort of question to which one cannot give an answer. What could one possibly use to explain what physical objects are? But philosophers actually do try to give some general sorts of accounts of what they are. They ask: Are physical objects just bundles of their properties? Or are they substances which have those properties? That is called the problem of substance or objecthood.
Here is another sort of question. We said that the apple has properties, like being red, being big, being juicy. How are properties different from objects? Notice, we say that things like apples have properties like redness. But apples and redness are different sorts of items, of things, of entities. One can pick up and touch an apple, but cannot pick up and touch redness, except perhaps in the sense that you can pick up and touch red things. So how can we best think about what properties are? This is called the problem of universals.
Here is another question about what physical objects are: when ingeneral can we say that physical objects comeintobeing and when they ceasetoexist? Surely the apple can change in many ways without ceasing to exist. It could get brown and rotten but it would still be that apple. But if someone ate it, it would not just have changed; it would no longer exist. So there are some metaphysical questions to be answered about the notions of identity, or being the same thing over time, and change.
This apple exists in space (it sits on a table in a room) and in time (it was not on the table a week ago and it will not be on the table a week from now). But what does this talk of space and time mean? Can we say, for example, that space is like an invisible three-dimensional grid in which the apple is located? Suppose the apple, and every other physical object in the universe, were to be entirely removed from existence: then would space, that "invisible grid," still exist? Some people say not -- they say that without physical objects, space would not exist, because space is the framework in which we understand how physical objects are related to each other. There are many other metaphysical questions to ask about space and time.
There are some other, very different sorts of problems in metaphysics. The apple is one sort of thing; now if Sally is in the room, and we say Sally has a mind, we are surely going to say that Sally's mind is a different sort of thing from the apple (if it is a sort of thing at all). I might say that my mind is immaterial, but the apple is a material object. Moreover, it sounds a little strange to say that Sally's mind is located in any particularplace; maybe we could say it is somewhere in the room; but the apple is obviously located in a particular place, namely on the middle of the table. It seems clear that minds are fundamentally different from physical bodies. But if so, how can something mental, like a decision to eat, cause a physical event to occur, like biting down on the apple? How are the mind and body causally interconnected if they are two totally different sorts of things? This is called the mind-body problem, which is now typically relegated to a philosophical subdiscipline called philosophy of mind. The mind-body problem is sometimes still considered part of metaphysics, however.