Brainfuck was first implemented and described by Urban Müller in 1993, attempting to design a language for which he could write the smallest possible compiler. The language was inspired by False. Müller decided to name his language brainfuck (with the initial letter in lower case, although it is now often capitalized). He uploaded the first Brainfuck compiler to Aminet in 1993. The program came with a readme file, which briefly described the language, and challenged the reader

Who can program anything useful with it? :)

Müller also included an interpreter and some quite elaborate examples

It is not known was Müller aware of or influenced by Böhm's language P'' (1964), of which brainfuck can be considered a minor variation.