equivalent positions

Equivalent positions are the full family of points that the crystal's symmetry ties together as identical. Start with one point, apply every symmetry operation of the space group, and each operation drops a copy somewhere; that whole set of copies are 'equivalent' — the same atom in the same kind of surroundings, indistinguishable by any physical measurement. Move to any one of them and the crystal around you looks exactly the same.

The International Tables list, for each space group, the coordinates of these equivalent positions — the general set first, then each special set. Take triclinic P-1: its only symmetry beyond translation is inversion, so the general equivalent positions are just (x, y, z) and (-x, -y, -z), a family of two. In P2_1/c the general family has four members. Crucially, these coordinate triplets ARE the space group's symmetry operations written out as a recipe — read (-x, -y, -z) as 'invert through the origin', read (x, 1/2-y, 1/2+z) as 'apply the c-glide'.

This is the machinery you actually use to build a model. Take each atom's asymmetric-unit coordinate, generate all its equivalent positions, then repeat by lattice translations, and the complete crystal appears. Every atom in an equivalent set scatters X-rays identically, bonds identically, and plays the same structural role — which is why you only ever have to determine one of them. (Beware a loose habit: some people say 'general positions' when they mean this general equivalent-position set.)