What do cosmologists mean when they talk about "the running of the spectral index"? All I know is that the spectral index is related to the CMB, and that it has to be slightly less than 1 to favor inflation.
The spectral index $n_s$ describes how the clumpiness of stuff varied on various scales just after cosmic inflation. The primordial power spectrum $P(k)$ of the fluctuations (where the wave number is $k=2\pi/\lambda$ with $\lambda$ being the physical scale), is 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 $k$, that is 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. Cherny et al. 2014).

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