wave interference
When two sets of ripples cross on a pond, something striking happens where they overlap: in some places the water heaves up higher than either ripple alone, and in others it goes flat and still. This is interference. Two waves passing through the same point at the same instant simply add: crest landing on crest gives a bigger crest (they reinforce), while crest landing on trough cancels to nothing (they annihilate). Nothing is created or destroyed — the energy is merely redistributed, piled up here and emptied out there.
What decides reinforce-or-cancel is the phase — how far one wave is shifted relative to the other, which comes down to the difference in the distance each has travelled, the path difference. If one wave has travelled a whole number of wavelengths farther than the other (path difference = 0, 1, 2, ... wavelengths), the two arrive in step and add: constructive interference. If it has travelled an extra half a wavelength (0.5, 1.5, 2.5, ...), they arrive exactly out of step and cancel: destructive interference. In between you get something partial. So a tiny change in geometry — a slightly different angle or spacing — flips brightness to darkness.
Diffraction is nothing more than this rule applied to the enormous number of wavelets scattered by an ordered array of atoms. In most directions the countless wavelets have every possible phase and wash each other out to darkness; only in the special directions where they are all in step does light survive and blaze out as a sharp beam. Everything else in this field — Bragg's law, the Laue condition, structure factors — is bookkeeping for one question: in which directions does the interference come out constructive?
Shine one colour of light through two narrow slits and a screen beyond shows bright and dark bands, not two blurs. The bright bands fall where the path from the two slits differs by 0, 1, 2, ... wavelengths; the dark bands where it differs by a half-integer. A crystal does the same with millions of scatterers, giving spots instead of bands.
Two-slit fringes are interference in miniature; a crystal is a three-dimensional grating obeying the very same rule.
Interference needs waves with a fixed phase relationship (coherent waves). This is why diffraction uses the coherent, elastically scattered part of the beam; randomly phased (incoherent) scattering just adds a smooth background.