a base-centered lattice
This time add just one extra lattice point, at the centre of one pair of opposite faces (say the top and bottom). That is a base-centered (or end-centered) lattice. Only two of the six faces get a central point, unlike face-centering which decorates all six.
The symbol tells you which pair: C-centred adds a point on the faces perpendicular to c (the ab faces), at (1/2,1/2,0); A-centred uses the bc faces, B-centred the ac faces. Counting: 8 corners times 1/8 + 2 faces times 1/2 = 1 + 1 = 2 lattice points per cell. Base-centering appears in the monoclinic and orthorhombic systems (for example, C-centred orthorhombic).
Base-centering is only distinct in lower-symmetry systems. In a cubic lattice you cannot base-center — singling out one axis would break the cubic symmetry (and it would just be describable as a smaller tetragonal cell). That is why the 14 Bravais lattices include base-centered monoclinic and orthorhombic but not base-centered cubic.
C-centered orthorhombic: points at the 8 corners plus one at the centre of each ab face (top and bottom), coordinate (1/2,1/2,0). Count = 8 times 1/8 + 2 times 1/2 = 2 points per cell. The a- and b-side faces stay bare — only the c-perpendicular pair is centered.
Base-centering decorates just one pair of opposite faces.
Whether a cell is called A-, B-, or C-centered depends only on which axis you label as the odd one out; merely relabeling axes can turn a B-centered cell into a C-centered one without changing any physics.