a lattice translation vector
If you stand on one lattice point and want to hop to another, the arrow describing that jump is a lattice translation vector. Because the pattern is periodic, sliding the whole crystal along such a vector leaves it looking exactly the same — an invisible move.
Any lattice translation can be written R = u a + v b + w c, where u, v, w are integers and a, b, c are the three basis vectors (the cell edges). The three shortest independent ones define the unit cell. For example, in a cubic lattice with edge a, the vector from a corner to the next corner along x is [100] = a; a face diagonal is [110] = a + b, of length a times sqrt(2).
These vectors are the translational symmetry of the crystal — the operations that map the lattice back onto itself. The fact that only whole-number combinations are allowed is exactly what forbids 5-fold rotational symmetry in a periodic crystal (the crystallographic restriction theorem). Written in square brackets, [uvw] denotes such a direction.
In an FCC lattice with cubic edge a, the vector from a corner to the nearest face-centre is (a/2)[110] = (a/2)(a + b), of length a/sqrt(2). Applying it slides the crystal onto itself. But (a/2)[100] is NOT a lattice translation — it lands you between lattice points, on no equivalent site.
Only integer combinations of the primitive vectors are true lattice translations.
Watch the basis: for a centered lattice the shortest true translations are combinations of the primitive vectors, which are shorter than the conventional cell edges. Whole-number multiples of the conventional edges are always translations, but they are not the only ones.