Understanding the relevance of frequency (as distinct from position) is crucial when one encounters resonant systems in nature. The reciprocal lattice is similarly useful when one encounters crystals. In the figure below, the direct lattice is on the left while the corresponding reciprocal lattice (frequency-space transform) is on the right. (Note: hit reload if both left and right models below don't load on the first pass.) The balls in the direct lattice correspond to atoms or molecules with spacing measured in distance units like meters or Angstroms, while the balls in the reciprocal lattice correspond to spots in a power spectrum (or in diffraction) with spacing measured in reciprocal-distance units e.g. reciprocal-Angstroms.
The left model shows a simple-cubic lattice a few unit-cells across. An "Ewald slice" (or diffraction pattern) of its corresponding reciprocal lattice is on the right. That bright central spot in the reciprocal space model is oft referred to as the "DC peak" or "unscattered beam". Also note that square or angle brackets are typically used for direct space vector indices, while round or curly brackets are used for reciprocal space (Miller) indices. The set of planes defined by the red atoms in direct space gives rise to the reciprocal lattice spots (perpendicular to that plane) in cyan on the right (best seen in <111>_view). What are the Miller indices of that plane? How about the Miller indices of the dark yellow spot also active in <111>_view? The buttons underneath allow you to modify the model. What happens to the reciprocal lattice when you add stuff in direct space? What if you shrink the spacing between scattering centers? Is the reciprocal lattice of a simple-cubic lattice also simple-cubic? How about the reciprocal lattice of a body-centered cubic crystal? How many three-fold symmetric zones do these lattices exhibit?