1. Scholastic usage. A general class of things containing subclasses called species. Whether a given class is considered a genus can be relative. If it contains subclasses but is itself contained in a more general class, it will relate to the subclasses as a genus but as a species to the more general class that contains it. E.g., “animal” is a genus in relation to “rational animal” and “non-rational animal” but a species in relation to “substance.”  In the technical terminology of classical logic, genericity is the very relation formed by the reasoning intellect (relatio rationis) between, for example, animal and man (defined as rational animal). This relation is correlative to the relation of specificity.  Thus, the genericity of animal in relation to man implies the specificity of man in relation to animal. 

Supreme Genera (also called Ultimate Genera).— The most general classes of things above which there are no classes that are more general. Aristotle and many scholastics consider the categories to be the supreme genera. 

2. Modern biological usage. The class of living things below “family” and above “species.”