LANG and LANGUAGE environment variable in Debian based systems

LANG contain the setting for all categories that are not directly set by a LC_* variable.

LC_ALL is used to override every LC_* and LANG and LANGUAGE. It should not be set in a normal user environment, but can be useful when you are writing a script that depend on the precise output of an internationalized command.

LANGUAGE is used to set messages languages (as LC_MESSAGES) to a multi-valued value, e.g., setting it to fr:de:en will use French messages where they exist; if not, it will use German messages, and will fall back to English if neither German nor French messages are available.

Have a look at the manpage locale(7): it describes that LANG is a fallback setting, while LC_ALL overrides all separate LC_* settings.

For reference, the locale system is GNU GetText, which has its full documentation available in the gettext-doc package (Debian/Ubuntu).

Alternatively, there is an online manual with authoritative and elaborate documentation of the LANG and LANGUAGE environment variables.