the path difference
Whenever two waves set out together, bounce off different places, and meet again, one has almost always travelled a little farther than the other. That extra distance is the path difference, and it is the master quantity of interference. Measured in wavelengths, it decides everything: a path difference of a whole number of wavelengths means the waves arrive back in step and brighten; a half-integer means they arrive opposed and cancel.
In the Bragg picture the path difference comes from two waves reflecting off neighbouring planes a distance d apart. Work the geometry out step by step: the wave that penetrates to the lower plane must go down into the crystal and back up, and the extra leg it travels compared with the upper wave is d sin theta on the way in plus d sin theta on the way out, for a total of 2 d sin theta. (Here theta is the glancing angle from the planes.) Constructive interference then demands 2 d sin theta = n lambda — which is Bragg's law, derived entirely from bookkeeping the path difference.
The same idea, stated more generally, is the heart of the Laue picture too: there the path difference is computed between waves scattered by two atoms separated by a lattice vector, and constructive interference requires it to be a whole number of wavelengths for every lattice vector at once. Whether you phrase it as planes (Bragg) or atoms (Laue), the physics is identical — line the path difference up to whole wavelengths and the array shines.
Take d = 2 angstrom and theta = 22.6 degrees. The path difference is 2 d sin theta = 2 x 2 x sin(22.6 degrees) = 2 x 2 x 0.385 = 1.54 angstrom — exactly one wavelength of copper radiation, so the waves reinforce and a first-order beam appears.
Path difference = 2 d sin theta; set it equal to one wavelength and you recover the first-order Bragg condition.
It is the path difference in wavelengths, not in absolute distance, that matters. The same 1.54 angstrom of extra path is one wavelength for copper X-rays but a fraction of a wavelength for a longer-wavelength probe, giving a different angle.