The language belonged to the people who lived on the peninsula, Celtic since they were called Britons. Cornwall was independent even in Anglo-Saxon time, and though some migrants from the peninsula flew in Bretagne, the population remained Celtic. In the 11th century it was conquered by Normans and since then Cornish people slowly began to die out together with their language which was replaced by English. Cornwall is not a community isolated by mountains such as Scotland and Wales - soon the language stopped defending, and in the 18th century the last speakers of Cornish disappeared, leaving just rich literature. It dates back to the 15th century and is represented mainly by poetry - there are a lot of verses from the 16th and 17th centuries written in Cornish.
The language is closely related to Breton, less relative to Welsh, Gaelic and Manx. Nowadays linguistic societies divide the language into three main versions or dialects called Kemmyn, Unified and Modern. Of them, Modern is based on the Middle and Old Cornish lexics and grammar, Kemmyn and Unified are very much the same with only difference in spelling. But the structure, the unique Celtic morphological structure, remains the same everywhere, and Cornish also has prepositional pronouns, initial mutations and verb-based syntax.
Cornish is certainly much more hard to study than English. But if I were Cornish, I would study it, and I believe people in Cornwall will give this beautiful tongue a chance for a revival.