the zone axis
Think of an open book: many pages, all sharing one spine. In a crystal, many different planes can share a common line — the direction that lies flat inside all of them. That shared line is the zone axis, and the collection of planes that share it is called a zone. Written as a direction [uvw], the zone axis is the crystal's spine that a whole set of planes hinges around.
There is a simple test for membership: a plane (hkl) belongs to the zone [uvw] if and only if hu + kv + lw = 0. And if you have two planes and want the axis of the zone they define, you take a kind of cross product of their indices. For planes (h1 k1 l1) and (h2 k2 l2) the zone axis is [uvw] with u = k1 l2 - k2 l1, v = l1 h2 - l2 h1, w = h1 k2 - h2 k1. For example the (100) and (010) planes both contain the vertical direction, and this recipe returns [001] — the axis they hinge on.
The idea earns its keep in electron microscopy. When you tilt a crystal so the electron beam runs straight down a zone axis [uvw], the diffraction pattern shows a neat symmetric array of spots from all the planes that contain that axis. Crystallographers call this being on the zone axis, and reading such patterns is how orientations and unit cells are worked out in the transmission electron microscope.
The planes (110) and (1-10) share the vertical line: applying the cross-product recipe gives u = 1(0)-(-1)(0)=0, v=0, w=1(-1)-1(1)=-2, which reduces to [001]. So both planes hinge around the c axis, and [001] is their zone axis.
The zone axis is the common direction shared by a set of planes — their spine, written [uvw].
A zone axis is a direction [uvw]; the planes in the zone are (hkl). They may share the same integers in cubic crystals, but conceptually a direction and a plane are different objects — keep the brackets straight.