homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
An $n$-functor is simply a functor between $n$-categories. Similarly, an $\infty$-functor is a functor between $\infty$-categories.
Of course, as the definition of $n$-category gets more complicated as $n$ increases, so does the appropriate definition of functor. This explains why one says ‘$n$-functor’ instead of simply ‘functor’ all along. On the other hand, anything that goes between $n$-categories, if it deserves to be called anything like ‘functor’ at all, will be an $n$-functor, so the prefix is not really necessary.
An $n$-natural transformation goes between $n$-functors, and there are things to go between those as well, etc. The most general concept is an $n$-$k$-transfor.