Cleavage planes follow specific lattice planes and can be classified with respect to their orientation relative to the main crystallographic axes and the crystal symmetry of a mineral.

Basal cleavage: a cleavage plane parallel to the basal face (or base) of a crystal (perpendicular to the c-axis). Very common in platy, lamellar minerals like micas, breaking as ‘sheets’.
Cubic cleavage: a cubic cleavage is characterized by three cleavage planes that intersect at 90°, producing small cubes. Considering the cubic symmetry of the minerals that show this kind of cleavage, the cubic cleavage is not produced by three different cleavage system but only by one, which is repeated by the symmetry of the mineral. The same happens for other three-dimensional cleavage systems (e.g. rhombohedral, octahedral, dodecahedral…). Cubic cleavage is common in cubic minerals like galena or halite.
Rhombohedral cleavage: similar to the cubic cleavage, but the angle between the three cleavage planes is different than 90°. Typical of trigonal and hexagonal minerals like calcite and dolomite.
Octahedral cleavage: it occurs in cubic minerals that crystallize as octahedrons. Four cleavage planes, repeated by the symmetry of the mineral, intersect producing small octahedrons. The classic example is fluorite.
Prismatic cleavage: occurs in prismatic minerals, whose cleavage planes are parallel to the long axis of the prism (the c-axis). Examples are amphiboles and pyroxenes.
Dodecahedral cleavage: some high symmetry minerals may have six cleavage planes that intersect in space separating small dodecahedrons (e.g. sphalerite).
Pinacoidal cleavage: it occurs in some prismatic minerals like barite, whose planes of weakness are oblique to both the basal and prismatic planes (i.e. pinacoidal plane).

How well cleavage planes are developed depends on the weakness of specific lattice planes relatively to the rest of the crystal. Some cleavages develop on planes that are extremely weak, while others along directions that are only slightly weaker compared to the rest of the crystal. As a result, cleavage planes range from well-developed to poorly-developed. Mineralogists developed a qualitative scheme to classify cleavage planes, based on the quality of cleavage:

Perfect: smooth cleavage surfaces.
Good: smooth cleavage surfaces with some rough asperities.
Poor: the cleavage surface is rough.
Indiscernible/indistinct: a cleavage that is so poor that is barely recognizable.
None: a mineral lacking cleavage (which can still fracture, though).