A hypothetical language family grouping two of the three families wholly indigenous to the Caucasus
Mountains: North-East Caucasian
(which includes Chechen
) and North-West Caucasian
. See my write-ups on those for details of them.
The two families have many similarities, though these might be because of thousands of years of contact rather than descent. Glottochronology shows that the two branches separated very deep in the past, if they are related.
They are definitely not relatable to the third family found only in the Caucasus, the Kartvelian (or South Caucasian) family of Georgian. Occasionally the name Caucasian is used for North Caucasian, but I think this is misleading, myself, because of all the other Caucasian languages that aren't in the North Caucasian macrofamily.
One possible wider connexion is the ancient Hurrian language, spoken in what is now Kurdistan. And a small minority of linguists would find affinities with a scattering of other languages across the world, from Basque to Chinese to Navaho, in the Dene-Caucasian hypothesis.