The name of the Cornish language in Cornish is 'Kernewek', or 'Kernuak' in some dialects. It is not a dialect of English, but a separate, older language. There is, however, also a Cornish dialect, which devolves from English. The Cornish language is related to Welsh, but closer to Breton - Cornish and Breton being mutually intelligible.
Cornish began to wane with the encroachment of English, and the introduction of the English Prayer Book. The last monoglot speaker of Cornish was Dolly Pentreath, who died in 1777. By this time, the language was already being revived by Henry Jenner, therefore has always been continually spoken - although not necessarily as a mother tongue. Cornish is not a dead language: there have always been people able to speak Cornish, and approximately 10% of people living in Cornwall (of whom 50% are actually Cornish, therefore about 20% of Cornish people) currently have knowledge of it, there are also many children who have Cornish as their mother tongue.
Cornish gets no support from the British government. Tuition in Cornish is denied (unlike Welsh), and the demand for this tuition is ignored. Cornish placenames are being systematically replaced with English ones, and the language constituting part of Cornish culture is being denied, contrary to the European charter for Regional and Minority Languages. This leads to widespread ignorance and discrimination against the Cornish, and Cornish speakers.
An example of the difference between Cornish and English are the numbers:
Un/Onen = One
Dew = Two
Try/Teyr = Three
Peswar/Peder = Four
Pymp = Five
Whegh = Six
Seyth = Seven
Eth = Eight
Naw = Nine
Deg = Ten
Kernow bys Vyken!