It is hoped, moreover, that a full account of the categories would be exhaustive in the sense that everything can be placed into at least one of the categories. Sometimes ontological category schemes have included nonexistent or even impossible objects; Meinong, who thought we can talk unobjectionably about nonexistent objects such as the golden mountain, was an ontologist.
For example, what it means to take the category physical object seriously as a category of being is to assert that the concept of physical objecthood cannot be reduced to or explicated in any other terms—not, for example, in terms of bundles of properties. In this way, as it turns out, very many controversies of ontology can be understood as controversies about exactly which categories should be regarded as the (fundamental, irreducible, primitive) categories.
Category came into use with Aristotle; one of his treatises is called the Categories (which can be found online, for example, here). [Somebody should list Aristotle's categories here.] Aristotle's particular list of categories is widely rejected nowadays, however, in part because the Aristotelian notion of substance has been widely rejected.
Philosophers have many differing views on what the fundamental categories of being are. In no particular order, here are at least some items that have been regarded as categories of being by someone or other:
Physical objects. Physical objects are beings; certainly they are said to be in the simple sense that they exist all around us. So a house is a being, a person's body is a being, a tree is a being, a cloud is a being, and so on. They are beings because, and in the sense that, they are physical objects. One might also call them bodies, or (physical) particulars, or concrete things, or maybe substances (but bear in mind the word 'substance' has some special philosophical meanings).
Minds. Minds—those the "parts" of us that think and perceive—are beings. Each of us, according to common sense anyway, has a mind, a mind that exists or has being. So each of our minds is a being. Of course, philosophers rarely just assume that minds are a different category of beings from physical objects. Some have thought that mind is in a different category (this is the view of dualism), while some have thought that concepts of the mental (e.g., our notion of the mind) can be reduced to physical concepts (this is the view of physicalism or materialism).
Classes. We can talk about all human beings, and the planets, and all engines as belonging to classes. Within the class of human beings is all of the human beings, or (in other words) the extension of the term 'human being'. In the class of planets would be Mercury, Venus, the Earth, etc.—and all the other planets that there might be in the universe. Classes, in addition to each of their members, are often taken to be beings. Surely we can say that in some sense, the class of planets is, or has being. Classes are usually taken to be abstract objects, like sets; 'class' is often regarded as equivalent in meaning to 'set'.
Properties. The redness of a red apple is a property. One could also call it an attribute of the apple. Very roughly put, a property is just a quality that describes an object. This will not do as a definition of the word 'property' because, like 'attribute', 'quality' is a near-synonym of 'property'. But these synonyms can at least help us to get a fix on the concept we are talking about. Whenever one talks about the size, color, weight, composition, and so forth, of an object, one talks about the object's properties. Some—though this is a point of severe contention in the problem of universals—believe that properties are beings; the redness of the apple is something that is.
Relations. An apple sitting on a table is in a relation to the table it sits on. So we can say that there is a relation between the apple and the table—namely, the relation of sitting-on. So—some say—we can say that that relation has being. Or how about this: the Washington Monument is taller than the White House. Being-taller-than is a relation between the two buildings. We can—it seems, anyway—say that that relation has being as well. (This too is a point of contention in the problem of universals.)
Properties, relations, and classes are supposed to be abstract, rather than concrete. Many philosophers say that properties and relations have an abstract existence, and that physical objects have a concrete existence. That, perhaps, is the paradigm case of a difference in ways in which items can be said to be, or to have being.