When we talk about the region commonly referred to as the Caucasus and especially the languages therein, we are in fact talking about several layered meanings. In common usage, the term Caucasian Language is sometimes used as a catch-all for every language within the region's geographical boundaries, but this is incorrect. In terms of what we would consider the indigenous language families in the region, there are properly three: Northwest, Northeast, and Southwest Caucasian. In addition to these language families, there are several other families represented within the region, however, despite their location, they are nevertheless still part of their respective lineages from elsewhere.
Northwest Caucasian - Two main branches and a third defunct, Abaza-Abkhaz and Adyghe(Circassian), and a branch for the functionally extinct Ubykh Language.
Northeast Caucasian - Two branches, the Nakh branch is Chechen, Ingush, and several others found sometimes in places like Dagestan and Georgia. The other branch is the Dagestani branch proper, which includes Avar, Lezgian, Tsezic and others.
Southwest Caucasian - Sometimes written in English as Kartvelian, this is the family where Georgian proper resides, although it is not the only member. Svan, Megrelian, and Laz fill out the rest of the family.
Hereafter are the languages representing families from elsewhere:
Russian, Armenian, Ossetian, Pontic Greek(in some parts of Georgia), and several others all represent the Indo-European family in its widest sense, Armenian is a language-family isolate while Ossetian, Talysh, Kurdish are Indo-Iranian and therefore are (distant, in some cases) relatives of Persian; Pontic Greek is true to its name and is categorized as part of the Attic-Ionic branch of Greek. Of course, Russian is part of the Slavic language family.
Turkic languages are incredibly common; Azeri in Azerbaijan, Turkmen, and Turkish (at the far southern extent of the region) are all Oghuz Turkic. Karachay-Balkar, Kumyk, and Urum are Kipchak-Cuman, while Nogai is from the Kipchak-Nogai branch.
At the northernmost extent of the Caucasus region is Kalmyk, an Oirat Mongolic language brought to the region by the Russians when they moved the Kalmyk people there several centuries ago.
In the region you will also find representatives of the Semitic language family in Assyrian and Bohtan Neo-Aramaic who have been there since fleeing genocide in World War 1.
This is where the distinction occurs, all these other languages are part of the historical and geographic context of this region rather than its linguistic roots.