A strict category is a category together with the structure of a set (or class) on its collection of objects; in particular, the objects can be compared for equality (not merely isomorphism). In contrast, a weak category is a category without such structure. Similarly, a strict functor is one which preserves equality of objects: F(x)=F(y) if x=y. (NB: this is not inherently size-related—both large and small categories can be either strict or weak.)
Note that most categories we care about are not strict.