The use of the term 'Caucasian' to mean what most of us would naturally call
white or
European is one of those puzzling quirks - you
know it's some antiquated bit of
ethnology, as if when we said
Semitic we were thinking of the Biblical
Noah's Ark story, descendants of
Shem the son of Noah - yet it persists; at least in US
cop-show parlance. It's not actually offensive to anyone, and terms like 'white' and 'European' are equally contentious, so I suppose it's seen as the least worst of a bad job.
Almost all the languages from Europe to India - English, German, French, Latin, Russian, Greek, Persian, Hindi, Bengali, etc etc - are grouped in an Indo-European language family. Some 5000 or 7000 years ago they were a single language spoken by the ancestors of most of us writing here. Debate is strong on when the ancestral Proto-Indo-European group lived, and where. But one of the candidate regions is the steppes of southern European Russia, north of the Caucasus mountains. Archaeology reveals a culture known as the kurgan culture after their burial mounds. Their technology (horses, ploughs, corn, etc.) and their environment (bears, beeches, snow, etc.) approximately coincide with words that seem to have been in the original Proto-Indo-European language. So not literally Caucasian, but close.
In modern linguistic terms, there is no such thing as a Caucasian group. Reference to "Caucasian languages" is so loose as to be meaningless. But the Caucasus is one of the regions of the earth with the richest diversity of languages. (Others are
New Guinea,
Cameroon, and the north-west coast of North America.) There are several
language families unique to the Caucasus.
To get rid of one possible misconception, if someone mentions Caucasian languages, they are not talking about the languages of the "Caucasian" = white race. Or if they are, you back away from them very carefully and look for a door.
Around the Caucasus, there are more familiar language groups. To the north of the mountains is Russia, and Russian is a Slavonic (or Slavic) language, part of the Indo-European family. To the south-east is Iran, and here we find Persian and Kurdish, which are Iranian languages, and also part of the great Indo-European family. And to the south-west is Turkey: Turkish is part of a different family called Turkic, and maybe part of a larger Altaic family.
To sum up, on the outside of the Caucasus region you have members of the Indo-European and Turkic language families. But inside the region, there is a phenomenal diversity.
There are now three independent countries in the Caucasus: Georgia, Armenia, and Azerbaijan. But being a mountainous region, it's incredibly fractious, and there are small territories that have been fighting their own little wars of independence: Chechnya, Abkhazia, Artsakh (Nagorny-Karabakh), and South Ossetia. That's the politics; how does this translate into language?
Armenia has its own unique language, Armenian. There is nothing else close to Armenian, but it's an isolated branch of Indo-European, so distantly related to the Slavonic branch that includes Russian, and the Iranian branch including Persian. The secessionist region of Artsakh inside Azerbaijan also speaks Armenian.
Azerbaijan has a language Azeri or Azerbaijani which is very close to Turkish. The Turks have only been in what we and they call Turkey for a thousand years. They are invaders from Central Asia, and their languages are very similar to Kazakh, Kyrgyz and so on: and Azeri is another one of those. It was planted in the Caucasus in historically recent times.
Russia has a number of autonomous republics in its southern flank, bounding the Caucasus: Adygea, Kabardino-Balkaria, Karachay-Cherkessia, North Ossetia - Alania, Ingushetia, Chechnya, Daghestan. Of these, the Ossetians (or Ossetes or Alans), who also live in the separatist South Ossetia region of neighbouring Georgia, speak a language of the Indo-European group, distantly related to Persian. Linguists now group it in the Iranian branch.
That disposes of all the languages in the Caucasus mountains that have any connexions with anything else on the planet Earth. What remains are lots and lots that simply cannot be reliably connected to anything outside. And there is not one group of uniquely Caucasian languages; not two groups; but three.
The South Caucasian group is basically Georgian, the majority language of Georgia. Other South Caucasian languages are pretty close, almost dialects: Mingrelian, Svan, Laz, spoken by not many people. The South Caucasian family is also called Kartvelian, from the native Georgian name for Georgia.
The North-East Caucasian group is several dozen, of which the most spoken is Chechen, but there are plenty more spoken in isolated valleys and villages.
The North-West Caucasian is four or five (one went extinct in 1992), including Abkhaz. These are remarkable for having lots of consonants and very few vowels.
The North-West and North-East families have some similarities that suggest they are part of a deeper North Caucasian family. But they are definitely not related to South Caucasian. The minority of linguists who like to use tenuous evidence to find deep world-wide connexions suggest that South Caucasian can ultimately be linked to Indo-European, while North Caucasian is part of the Dene-Caucasian superfamily. The jury is definitely out on all this. 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.