A hypersurface with constant (anisotropic) mean curvature is said to be stable if the second variation of the area (the anisotropic surface energy) is nonnegative for all "volume"-preserving variations satisfying given boundary conditions. A useful criterion for the stability is given by using an elliptic operator associated with the second variation. In this talk, we apply it to the problem of the structure of the space of hypersurfaces with constant (anisotropic) mean curvature with prescribed boundary conditions. We will give general methods and their applications to several concrete examples which may be interesting from both mathematical and physical point of view.