Also spelled Chagatai, Chaghatay, etc., sometimes simply referred to as Turki or Türki. This extinct Eastern Turkic language is considered the precursor to modern Uzbek and Uyghur. More importantly, it was a classical language of literature and statesmanship in territory stretching from Mongol-controlled Central Europe in the west, to Xinjiang in the east, and from as far north as Mongolia to as far south as Mughal India, in use (alongside other languages) from the 13th-19th centuries. As such, it has a very rich body of literature including poetry, treatises on various topics (ie. Muhakamat al-Lughatayn), and memoirs (ie. the memoir of Mughal emperor Babur). For that reason, I've had a longstanding interest in learning to read Chagatay; I thought I'd start a thread to stir up some interest in it.
ABOUT CHAGATAY
Chaghatay (Encyclopedia Iranica)
Chaghatay Language and Literature (Encyclopedia Iranica)
The Wikipedia article is also not bad. Some of the editions of the Wikipedia article written in other Turkic languages (ie. Turkish, Uyghur) are interesting in that they feature comparisons between Chagatay and modern Turkic languages, and also delve into the orthography.
LEARNING RESOURCES
In English, Eckmann's Chagatay manual and Bodrogligeti's A Grammar of Chagatay are probably the best sources available and are fairly comprehensive. They can both be found for free on e-book filesharing websites.
I have come across many other resources for learning Chagatay written in Russian, German, and Turkish, but I can't comment on their quality. It is taught in many Turkish universities.