A recent paper (A dusty compact object bridging galaxies and quasars at cosmic dawn) describes one of their results as the "Effective radius of [CII] line" (Extended Data Table 2, p38):
I know that [CII] denotes the forbidden emission line of the second ionised state of carbon, but what do they mean when they say that its effective radius is "$1.4 \pm 0.2$ kiloparsecs"? Is this referring to the effective radius of the region of the quasar emitting that line?
As the surface brightness (SB) of extended objects does not reach zero at some well-defined radius, we need a measure to be able to compare various objects. Probably the most used measure is the effective radius $r_e$, which is also called the half-light radius ($r_{1/2}$), because it is the radius within which half of the total light is emitted.
Other useful definitions exist, e.g. the radius enclosing 90% of the light ($r_{90}$), or the Petrosian radius ($r_p$).
Using such definitions makes it less important exactly when the SB reaches zero, as the outermost part don't contribute much to the total light anyway.
As Lucas comments, the regions emitting light at various wavelengths do not necessarily share the same morphology, so looking at different wavelengths yields different radii, which can tell you something physical about the object.
An example is comparing the optical or UV radius with that of Lyman $\alpha$. While the former is emitted mostly by centrally located stars, Lyman $\alpha$ scatters on neutral hydrogen (HI), and since galaxies are often enshrouded in HI halos, they will look bigger when observed in Lyman $\alpha$.
In order to measure the size of an object which is not too odd-shaped, usually some functional form is fitted to the SB. In the case of the paper you link to (Fujimoto et al. 2022), they fit a 2D elliptical Gaussian to both the 1.3 mm continuum and to the [CII] 158 µm line. While the latter returns a best-fit radius of $r_e=1.4\pm0.21\,\mathrm{kpc}$, the former doesn't converge, indicating that the continuum is not spatially resolved. Whether or not it's a quasar or a very compact starburst is a bit up to interpretation though.
In other words, to answer your specific question
Is this referring to the effective radius of the region of the quasar emitting that line?
the 1.4 kpc [CII] region is not the quasar itself (which is a point source). Rather, it is the central parts of the host galaxy, ionized by a quasar and/or general star formation.
The first-author is actually sitting in the office next to me; I'll make sure to tell him to give the proper definitions next time ;-)
