a ferroelectric domain

You might expect a freshly grown ferroelectric crystal to be polarized uniformly, all its dipoles pointing one way. It is not. Instead it splits itself into many small regions, each internally polarized along one of the allowed directions but with neighbouring regions pointing differently — like a field of grass where patches lean north, others east, others south. Each uniformly-polarized patch is a ferroelectric domain, and the thin boundary between two of them is a domain wall.

The crystal breaks into domains to lower its own energy. A single uniformly-polarized block would leave uncompensated charge on its faces, creating a strong depolarizing field that costs a great deal of electrostatic energy; splitting into oppositely-pointing domains lets those fields largely cancel. The distortion that accompanies polarization also stores elastic strain, which domains relieve. The walls come in types set by the crystal: a 180-degree wall separates domains pointing exactly head-to-tail, while a 90-degree wall in tetragonal barium titanate separates polarizations at right angles and also accommodates the shape change. Crucially, walls can move: apply a field and the domains oriented favourably grow at the expense of the others as their walls sweep sideways. That wall motion — 'poling' when it is deliberate and near-permanent — is also a large part of the material's dielectric and piezoelectric response, the so-called extrinsic contribution on top of the intrinsic lattice response.

Domains are why an as-fired ferroelectric ceramic shows almost no net external effect: its randomly oriented domains cancel, giving a piece that is ferroelectric internally yet dead on the outside. To wake it up you pole it — apply a strong DC field, usually while warm, to grow and align the domains into a net polarization, and only then is it a working piezoelectric or pyroelectric. An honest limit: in a polycrystalline ceramic each grain has its own crystal axes pointing at random, so even perfect poling within each grain cannot make all grains agree; a poled ceramic therefore reaches only a fraction of the single-crystal spontaneous polarization, and domain-wall motion, while useful, is also a source of loss, nonlinearity, and slow aging.