Azeri is a language spoken in Azerbaijan, Iran, Turkey, and Georgia. It is closely related to Turkish, and Turks and Azeris can generally understand each other without special accommodation. Indeed, there are dialects of Turkish spoken in eastern Anatolia that are closer to Azeri than they are to standard Istanbul Turkish. So it would not be amiss to say that Azeri is a dialect of Turkish. We might even go so far as to simply identify Azeri with Turkish. This is done in Iran, the northern part of which is home to more than half of the world’s Azeri speakers.
When Azeri and Turkish are taken to be distinct, it’s because Azeris and Turks are culturally and historically distinct. Initially they were just one big bunch of Turks, but in the 1500s the eastern part of that group was conquered by the Persians and then converted to Shia Islam, whereas before they had been, as Anatolian Turks today still are, Sunnis. So we might take Azeri to be a sociolect, namely, Turkish as spoken by Shias.
After being separated from the Anatolian Turks, the Azeris would be further split by the Russo-Persian Wars of the nineteenth century. The Treaties of Gulistan and Turkmenchay gave the Russian Empire the northern third of the land inhabited by Azeris (modern Azerbaijan). The remaining two thirds remained with Persia, resulting in two distinctive varieties of Azeri, Northern Azeri being heavily influenced by Russian and Southern Azeri being heavily influenced by Persian.
In Georgia, Azeri is widely spoken in the southern region of Borchali (Azeri: Borçalı). Borchali belongs to the Kvemo Kartli (Lower Kartli) administrative division and sits right about where the borders of Georgia, Azerbaijan, and Armenia meet (and just south of Tbilisi as well). Azeris are the largest minority group in Georgia, and although their population has decreased since Georgian independence, their percentage of the population of Georgia as a whole has actually gone up, to around 7%. Georgian is not well-known among Georgian Azeris, but, as usual, Russian is.
Azeris, by the way, are not the only Turks to have lived in Georgia. For several centuries, the Meskheti (or Samtskhe) region of Georgia was home to the Meskhetian Turks (also known as Ahiska Turks, after the Turkish name for the town Akhaltsikhe). Their dialect was intermediate between Istanbul Turkish and Azeri, and in fact they were classified at various times as either Turks or Azeris (according to the needs of whoever was in charge of the census at the time). Like their Turkic cousins the Balkars and Karachays, the Meskhetian Turks were deported to Central Asia in 1944. Unlike the North Caucasians, however, there wasn’t even an allegation of treason. The Soviets were planning for the possibility of war with Turkey, and they simply didn’t want any Turks near the border, so the Meskhetians had to go. Also unlike the North Caucasians, they were never allowed to return. They’re still refugees today, and some are starting to trickle into the United States.
At the turn of the twentieth century, Azeri and Turkish (like the other languages of Islamic lands) were written in Arabic script. Iranian Azeris never stopped using it, but the Azeris of Azerbaijan switched to the Latin alphabet in 1923. This switch is usually interpreted as part of an attempt by Soviet authorities to reduce the influence of Islam in Azerbaijan, but there was also intellectual support for Latinization on independent grounds. In 1929, Turkey too adopted the Latin alphabet (one of Atatürk’s Reforms). This prompted Soviet authorities, hoping to sever ties between Turkey and Azerbaijan, to convert Azerbaijan to the Cyrillic alphabet ten years later. So Azerbaijan went through three writing systems in less than twenty years. Since independence, Azerbaijan has mostly switched back to Latin, though Cyrillic can still be found.
Note on Terminology: “Azeri” vs. “Azerbaijani”
I’ve been using “Azeri”, but “Azerbaijani” is somewhat more common and is recommended by some. I don’t have terribly strong feelings about the issue. My reasons for choosing “Azeri” (besides following the style of certain scholars) are 1) most Azeris don’t live in Azerbaijan, so I would prefer to reserve “Azerbaijani” for referring to things related to the state of Azerbaijan, as distinct from general Azeri things; and 2) “Azerbaijani” is an ugly and cumbersome adjective, so I would prefer not to use it if I can avoid it. A third argument one could give (which does not convince me at all) is that Azerbaijan is not the real name of the region in the first place.