The name is a filing accident. When Aristotle's works were catalogued, the books dealing with being and first causes were placed after those on physics, and got labelled ta meta ta physika — the ones after the Physics. The word has since acquired connotations of the supernatural that have nothing to do with the subject, which concerns the structure of reality, not spirits.
The clearest way to locate metaphysics is by what it asks relative to science. A physicist investigates which events cause which. A metaphysician asks what causation is — whether it is a real relation in the world or a pattern we impose, whether causes necessitate their effects or merely make them probable. Science establishes that objects persist; metaphysics asks what persistence consists in, given that the matter is constantly replaced. These questions are presupposed by empirical work rather than settled by it.
Its central topics recur throughout philosophy. Ontology, the study of what exists, asks whether numbers, properties and possibilities are real things or useful fictions. Time raises whether past and future exist or only the present, and whether time's passage is real or a feature of how we experience it. Identity asks what makes something the same thing across change — the ship of Theseus problem, and its personal version. Modality concerns possibility and necessity: what does it mean to say something could have been otherwise?
The subject has been declared dead more than once. Hume proposed consigning metaphysical works to the flames as containing nothing but sophistry. Kant argued that reason inevitably generates metaphysical questions it cannot answer, and restricted legitimate knowledge to appearances. The logical positivists held that metaphysical statements are literally meaningless, since no observation could verify them — an argument that famously failed its own test, since the verification principle is not itself verifiable. Metaphysics survived all three, partly because the questions reassert themselves the moment anyone tries to say precisely what a scientific theory is claiming.