In higher category theory the term weak -categories is essentially synonymous with infinity-category in the fully general sense of (infinity,infinity)-category. The terms “-category” and “-category” originate in different schools and their choice of use is mostly a matter of the preference of individual authors.
One slight difference is that “-category” usually implies a “weak” (fully general) notion, while in addition to weak -categories there are also strict ones. Another difference is that definitions of weak -categories tend to be algebraic instead of geometric (accordingly typically the central open question is whether a definition really satisfies the homotopy hypothesis), though some definitions of weak -categories are geometry (for instance some flavors of definition of opetopic omega-category).
The following are examples for proposals of definitions of weak -categories.
See the general references at higher category theory.