What does the number 256 in pairing curve BN256 indicate?
It's the size of the prime number of the underlying field in G1, G2 and GT.

In BN256, G1 is $E(\mathrm{GF}(p))$, G2 is a subgroup of $E(\mathrm{GF}(p^{12}))$ (or $E'(\mathrm{GF}(p^{2}))$ when using a twist) and GT is a subgroup of $\mathrm{GF}(p^{12})$.

Elements of G1 requires the same number of bits as $p$ for each elliptic curve point coordinate. Note that not all prime-friendly curves support cofactor 1, which means that in some cases we need a larger prime for a particular group order. (This is not the case in BN curves).

Elements of G2 require the same as $p^{k}$ for each elliptic curve point coordinate, where $k$ is the embedding degree of the curve. When using twists, we can reduce this by 2, 3, 4 ou 6 depending on the curve. BN curves have embedding degree 12 and support sextic twists, so we can use elements with the same size as $p^{12/6} = p^2$.

Elements of GT require the same size as $p^{k}$.