[1] This means that both the objects and the morphisms of C and D stand in a one-to-one correspondence to each other.
Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms.
Isomorphism of categories is a very strong condition and rarely satisfied in practice.
As is true for any notion of isomorphism, we have the following general properties formally similar to an equivalence relation: A functor F : C → D yields an isomorphism of categories if and only if it is bijective on objects and on morphism sets.
[1] This criterion can be convenient as it avoids the need to construct the inverse functor G.