an aperiodic crystal
For most of crystallography, 'crystal' and 'periodic' were treated as synonyms: a crystal was a motif stamped over and over on a repeating lattice. An aperiodic crystal breaks that link. It is a solid that is genuinely ordered — its atoms sit in fixed, long-range, predictable relationships and it gives a sharp discrete diffraction pattern — but it has NO three-dimensional lattice repeat. Aperiodic crystal is the umbrella term for all such solids: quasicrystals, incommensurate structures, and modulated structures all live under it.
The defining test is diffraction, not appearance. Any ordinary crystal diffracts into sharp spots that you can label with three whole numbers (h, k, l), one per lattice direction. An aperiodic crystal ALSO diffracts into sharp spots — that sharpness is the proof of true long-range order — but you cannot index every spot with just three integers. Quasicrystals need more integers (an icosahedral one needs six); incommensurate and modulated structures need a main set of three plus extra 'satellite' integers for a second periodicity that does not fit the first. In every case the pattern is discrete (not the diffuse halo of a glass), so the material is a crystal in the modern sense, just not a periodic one.
This is why the International Union of Crystallography rewrote its definition in the 1990s: a crystal is now any solid having an essentially discrete diffraction diagram, full stop — periodicity is no longer part of the definition. Aperiodic crystals therefore matter as the whole reason the field's foundational concept had to change. They are described rigorously by embedding them as a periodic slice of a higher-dimensional lattice (superspace), which restores a kind of periodicity in more than three dimensions and lets crystallographers apply their usual machinery to solids that, in ordinary space, never repeat.
A powder pattern of an ordinary crystal indexes fully with three integers (hkl). A modulated aperiodic crystal indexes as (hklm): three main integers plus a fourth for extra satellite peaks sitting near each main peak. The need for that fourth number, while the spots stay sharp, is the operational proof it is aperiodic yet still a crystal.
An aperiodic crystal: sharp (discrete) diffraction but no 3D lattice repeat — the modern definition of 'crystal' covers it.
Aperiodic is not the same as amorphous. An amorphous glass gives diffuse haloes (short-range order only); an aperiodic crystal gives SHARP spots (genuine long-range order) and simply lacks a repeating unit cell.