A category is discrete if it is both a groupoid and a preorder. That is, every morphism should be invertible, any two parallel morphisms should be equal. The idea is that in a discrete category, no two distinct (nonisomorphic) objects are connectable by any path (morphism), and an object connects to itself only through its identity morphism.
Often one also assumes that a discrete category is skeletal; a category is both discrete and skeletal if and only if it contains only identity morphisms. However, this definition is evil, because it states that objects (the source and target of the identity morphism in question) are equal; it is cleaner to separate the discreteness from the skeletality.
A (small) discrete category may be identified with its set of isomorphism classes. Conversely, given a collection of objects, the discrete category over is the category with as its collection of objects and only identity morphisms.
If is a category enriched or internal to topological spaces, then there is another completely different meaning of discrete: that the topology on the arrows (and the objects, in the internal case) is the discrete topology.
This is especially confusing if one extends the use of “discrete category over ” to the case of internal categories, when is an object of some ambient category. With this usage, if is a topological space, then the “discrete internal category over ” in Top will not be discrete in the topological sense: it still remembers the topology on that space.