lattice centering
Sometimes the smallest cell that shows the crystal's true symmetry has extra lattice points not just at its corners but also in the middle of the body or on the faces. Adding those extra points is called centering. It is the reason we accept a larger-than-primitive cell: to keep the box's shape matching the symmetry.
The named centering types are given by a letter in the lattice's short symbol: P (primitive, corners only), I (body-centred, one extra point at the cell centre), F (face-centred, one extra at the centre of each of the 6 faces), and A, B, or C (base-centred, one extra at the centre of one pair of opposite faces), plus R (rhombohedral). For example, a C-centred cell adds a point at the centre of the ab faces, at fractional coordinate (1/2, 1/2, 0).
Centering has a direct diffraction fingerprint: the extra lattice points cause systematic absences — whole families of reflections vanish. Body-centering (I) kills reflections where h+k+l is odd; face-centering (F) kills those where h, k, l have mixed parity. So diffraction tells you the centering before you even know the atoms.
Take a primitive tetragonal cell and add a point at its body centre: you might think you have made a new lattice, but sometimes centering yields nothing new. Body-centered and face-centered tetragonal, for instance, are the same set of lattices seen through differently sized boxes — which is why only some centerings survive into the final list of 14.
Centering only counts when it produces a genuinely new lattice.
The centering letter is part of the Bravais and space-group symbol (for example the F in Fm-3m). It is a property of the lattice, decided by symmetry — not something you are free to add or remove at will.