n-category = (n,n)-category
n-groupoid = (n,0)-category
A -category is any of several concepts that generalize -categories one step in higher category theory. The original notion is that of a globular strict 3-category, but the one most often used here is that of a tricategory. The concept generalizes to -categories.
Fix a meaning of -category, however weak or strict you wish. Then a -category is an -category such that every 4-morphism is an equivalence, and all parallel pairs of -morphisms are equivalent for . Thus, up to equivalence, there is no point in mentioning anything beyond -morphisms, except whether two given parallel -morphisms are equivalent. This definition may give a concept more general than your preferred definition of -category, but it will be equivalent; basically, you may have to rephrase equivalence of -morphisms as equality.