the slip system
If slip is atoms sliding, a slip system tells you exactly which way they slide: a specific plane they slide along, paired with a specific direction they slide in. Think of pushing a stack of paper, which slides most easily along the flat faces (the plane) and in the direction you push (the direction). A crystal has favourite planes and directions built into its atomic packing, and slip happens on those.
The favourites are the densest ones: slip prefers the most closely packed plane (atoms most tightly tiled, so the plane is smoothest and spaced widest from the next) and the most closely packed direction within it (the shortest atomic step). In a face-centered-cubic (FCC) metal the close-packed planes are the {111} family and the directions are the <110> set; combining 4 distinct planes with 3 directions each gives 12 slip systems, all close-packed. That abundance of easy paths is why FCC metals like copper, aluminum, gold, and austenitic stainless steel are so ductile.
Crystal structure therefore governs ductility. Body-centered-cubic (BCC) metals like iron have up to 48 slip systems but none are truly close-packed, so slip needs more stress and gets much harder when cold, part of why steel can turn brittle in the cold. Hexagonal-close-packed (HCP) metals like magnesium and zinc have close-packed planes but only a few independent slip systems, so they are relatively brittle and hard to form at room temperature.
FCC aluminum has 12 close-packed slip systems and can be rolled into kitchen foil; HCP magnesium has far fewer active ones at room temperature and tends to crack when bent cold, which is why magnesium is usually shaped hot.
Same idea (slip), very different formability, because the number of easy slip systems is set by the crystal structure.
A material needs at least five independent slip systems to deform plastically into any general shape (the von Mises criterion); FCC easily clears this bar, HCP often does not, which is the real reason HCP metals are less ductile.