cosmology - What do cosmologists mean when they talk about "the running of the spectral index"?

The spectral index $n_s$ describes how the clumpiness of stuff varies on various scales. If you observe the CMB and take its power spectrum $P$, this is a function of the wave number $k$ (where $k=2pi/lambda$ with $lambda$ being the physical scale), predicted by many inflationary models to be:
$$P(k) propto k^{n_s-1}.$$
If $n_s=1$, the fluctuations are scale invariant.

If $n_s$ is not a constant, but instead changes with $k$, i.e. if
$$frac{dn_s}{d ln k} neq 0,$$
it is called a "running spectral index". And in fact it seems that that $n_s$ does chance with $k$, see e.g. here.

The term "the running of the spectral index" refers to the quantity $dn_s,/,dln k$.