Celsius - Centigrade
The two names Celsius and Centigrade have been used for the same scale for measuring temperature. This is the scale which takes the freezing point of pure water as 0° and the boiling point as 100°. The abbreviation for both names is simply C, or °C (for 'degrees Celsius').
Celsius should be the preferred name in academic circles, and certainly is in all the exact sciences. It has been the proper word to use since it was adopted by the International Standards Bureau in 1948, in the 9th CGPM (Conférence générale des poids et mesures) and the CIPM (Comité international des poids et mesures). It was chosen in part because centigrade, which is the usual term in colloquial English, is ambiguous: some cultures, and several languages, use 'centigrade' to mean degrees in other scales. (It has been used for "one-hundredth of a right angle", for example.)
Do not use
centigrade for academic work to mean temperature, and try to remember when you use the abbreviation °C that it stands for degree Celsius, not 'degree centigrade'.
The traditional scale in English-speaking countries for temperature was the Fahrenheit scale. This is now out of use in formal academic writing.