When $27$ cubes are assembled, one of the cubes fills the center, leaving $26$ cubes visible on the outside. How many "corner" cubes are there (with 3 sides visible), how many are "edge" cubes (with 2 sides visible) and how many are "center" cubes (with 1 side visible)?

Each of the smaller cubes in the $3$ by $3$ by $3$ configuration that is touching a vertex of the larger cube has $3$ sides visible; there are $8$ vertices of the larger cube, so there are $8$ corner cubes.
Each of the $6$ sides of the larger cube has $4$ "edge" cubes as marked where $2$ sides are visible.
Each of the $6$ sides of the larger cube has $1$ "center" cube in the middle with only $1$ side visible.