Ductility occurs in materials where valence shell electrons are not rigidly bound and are thus shared. These electrons atoms to slide past each other without breaking. It occurs under tension.
Brittleness is the tendency of a material to fail under stress, usually tension, rather than deform. It's not normally applied to shear. Generally, if there is little or no plastic deformation before failure, the material is said to be brittle.
Malleability is similar, but not identical, as it involves deformation under compression. Gold, for example, is both ductile and malleable, but lead is only malleable.