Brainfuck
Brainfuck is an esoteric programming language created in 1993 by Urban Müller.

Notable for its extreme minimalism, the language consists of only eight simple commands, a data pointer and an instruction pointer. While it is fully Turing complete, it is not intended for practical use, but to challenge and amuse programmers. Brainfuck requires one to break commands into microscopic steps.

The language's name is a reference to the slang term brainfuck, which refers to things so complicated or unusual that they exceed the limits of one's understanding, as it was not meant or made for designing actual software but to challenge the boundaries of computer programming.

History 
Müller designed Brainfuck with the goal of implementing the smallest possible compiler, inspired by the 1024-byte compiler for the FALSE programming language. Müller's original compiler was implemented in machine language and compiled to a binary with a size of 296 bytes. 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 examples. A second version of the compiler used only 240 bytes.

P′′ 
Except for its two I/O commands, Brainfuck is a minor variation of the formal programming language P′′ created by Corrado Böhm in 1964, which is explicitly based on the Turing machine. In fact, using six symbols equivalent to the respective Brainfuck commands +, -, <, >, [, ], Böhm provided an explicit program for each of the basic functions that together serve to compute any computable function. So the first "Brainfuck" programs appear in Böhm's 1964 paper – and they were sufficient to prove Turing completeness.

Language design 
The language consists of eight commands. A brainfuck program is a sequence of these commands, possibly interspersed with other characters (which are ignored). The commands are executed sequentially, with some exceptions: an instruction pointer begins at the first command, and each command it points to is executed, after which it normally moves forward to the next command. The program terminates when the instruction pointer moves past the last command.

The brainfuck language uses a simple machine model consisting of the program and instruction pointer, as well as a one-dimensional array of at least 30,000 byte cells initialized to zero; a movable data pointer (initialized to point to the leftmost byte of the array); and two streams of bytes for input and output (most often connected to a keyboard and a monitor respectively, and using the ASCII character encoding).

The eight language commands each consist of a single character:

[ and ] match as parentheses usually do: each [ matches exactly one ] and vice versa, the [ comes first, and there can be no unmatched [ or ] between the two.

As the name suggests, Brainfuck programs tend to be difficult to comprehend. This is partly because any mildly complex task requires a long sequence of commands and partly because the program's text gives no direct indications of the program's state. These, as well as Brainfuck's inefficiency and its limited input/output capabilities, are some of the reasons it is not used for serious programming. Nonetheless, like any Turing complete language, Brainfuck is theoretically capable of computing any computable function or simulating any other computational model, if given access to an unlimited amount of memory. A variety of Brainfuck programs have been written. Although Brainfuck programs, especially complicated ones, are difficult to write, it is quite trivial to write an interpreter for Brainfuck in a more typical language such as C due to its simplicity. There even exist Brainfuck interpreters written in the Brainfuck language itself.

Brainfuck is an example of a so-called Turing tarpit: It can be used to write any program, but it is not practical to do so, because Brainfuck provides so little abstraction that the programs get very long or complicated.

Examples

Adding two values 
As a first, simple example, the following code snippet will add the current cell's value to the next cell: Each time the loop is executed, the current cell is decremented, the data pointer moves to the right, that next cell is incremented, and the data pointer moves left again. This sequence is repeated until the starting cell is 0.

[->+<]

This can be incorporated into a simple addition program as follows:

++       Cell c0 = 2
> +++++  Cell c1 = 5

[        Start your loops with your cell pointer on the loop counter (c1 in our case)
< +      Add 1 to c0
> -      Subtract 1 from c1
]        End your loops with the cell pointer on the loop counter

At this point our program has added 5 to 2 leaving 7 in c0 and 0 in c1
but we cannot output this value to the terminal since it is not ASCII encoded

To display the ASCII character "7" we must add 48 to the value 7
We use a loop to compute 48 = 6 * 8

++++ ++++  c1 = 8 and this will be our loop counter again
[
< +++ +++  Add 6 to c0
> -        Subtract 1 from c1
]
< .        Print out c0 which has the value 55 which translates to "7"!

Hello World! 
The following program prints "Hello World!" and a newline to the screen:

[ This program prints "Hello World!" and a newline to the screen, its
  length is 106 active command characters. [It is not the shortest.]

  This loop is an "initial comment loop", a simple way of adding a comment
  to a BF program such that you don't have to worry about any command
  characters. Any ".", ",", "+", "-", "<" and ">" characters are simply
  ignored, the "[" and "]" characters just have to be balanced. This
  loop and the commands it contains are ignored because the current cell
  defaults to a value of 0; the 0 value causes this loop to be skipped.
]
++++++++               Set Cell #0 to 8
[
    >++++               Add 4 to Cell #1; this will always set Cell #1 to 4
    [                   as the cell will be cleared by the loop
        >++             Add 2 to Cell #2
        >+++            Add 3 to Cell #3
        >+++            Add 3 to Cell #4
        >+              Add 1 to Cell #5
        <<<<-           Decrement the loop counter in Cell #1
    ]                   Loop until Cell #1 is zero; number of iterations is 4
    >+                  Add 1 to Cell #2
    >+                  Add 1 to Cell #3
    >-                  Subtract 1 from Cell #4
    >>+                 Add 1 to Cell #6
    [<]                 Move back to the first zero cell you find; this will
                        be Cell #1 which was cleared by the previous loop
    <-                  Decrement the loop Counter in Cell #0
]                       Loop until Cell #0 is zero; number of iterations is 8

The result of this is:
Cell no :   0   1   2   3   4   5   6
Contents:   0   0  72 104  88  32   8
Pointer :   ^

>>.                     Cell #2 has value 72 which is 'H'
>---.                   Subtract 3 from Cell #3 to get 101 which is 'e'
+++++++..+++.           Likewise for 'llo' from Cell #3
>>.                     Cell #5 is 32 for the space
<-.                     Subtract 1 from Cell #4 for 87 to give a 'W'
<.                      Cell #3 was set to 'o' from the end of 'Hello'
+++.------.--------.    Cell #3 for 'rl' and 'd'
>>+.                    Add 1 to Cell #5 gives us an exclamation point
>++.                    And finally a newline from Cell #6

For "readability", this code has been spread across many lines, and blanks and comments have been added. Brainfuck ignores all characters except the eight commands +-<>[],. so no special syntax for comments is needed (as long as the comments do not contain the command characters). The code could just as well have been written as:
++++++++[>++++[>++>+++>+++>+<<<<-]>+>+>->>+[<]<-]>>.>---.+++++++..+++.>>.<-.<.+++.------.--------.>>+.>++.

Another example of a code golfed version that prints Hello, World!:

+[-->-[>>+>-----<<]<--<---]>-.>>>+.>>..+++[.>]<<<<.+++.------.<<-.>>>>+.

ROT13 
This program enciphers its input with the ROT13 cipher. To do this, it must map characters A-M (ASCII 65–77) to N-Z (78-90), and vice versa. Also it must map a-m (97-109) to n-z (110-122) and vice versa. It must map all other characters to themselves; it reads characters one at a time and outputs their enciphered equivalents until it reads an EOF (here assumed to be represented as either -1 or "no change"), at which point the program terminates.

-,+[                         Read first character and start outer character reading loop
    -[                       Skip forward if character is 0
        >>++++[>++++++++<-]  Set up divisor (32) for division loop
                               (MEMORY LAYOUT: dividend copy remainder divisor quotient zero zero)
        <+<-[                Set up dividend (x minus 1) and enter division loop
            >+>+>-[>>>]      Increase copy and remainder / reduce divisor / Normal case: skip forward
            <[[>+<-]>>+>]    Special case: move remainder back to divisor and increase quotient
            <<<<<-           Decrement dividend
        ]                    End division loop
    ]>>>[-]+                 End skip loop; zero former divisor and reuse space for a flag
    >--[-[<->+++[-]]]<[         Zero that flag unless quotient was 2 or 3; zero quotient; check flag
        ++++++++++++<[       If flag then set up divisor (13) for second division loop
                               (MEMORY LAYOUT: zero copy dividend divisor remainder quotient zero zero)
            >-[>+>>]         Reduce divisor; Normal case: increase remainder
            >[+[<+>-]>+>>]   Special case: increase remainder / move it back to divisor / increase quotient
            <<<<<-           Decrease dividend
        ]                    End division loop
        >>[<+>-]             Add remainder back to divisor to get a useful 13
        >[                   Skip forward if quotient was 0
            -[               Decrement quotient and skip forward if quotient was 1
                -<<[-]>>     Zero quotient and divisor if quotient was 2
            ]<<[<<->>-]>>    Zero divisor and subtract 13 from copy if quotient was 1
        ]<<[<<+>>-]          Zero divisor and add 13 to copy if quotient was 0
    ]                        End outer skip loop (jump to here if ((character minus 1)/32) was not 2 or 3)
    <[-]                     Clear remainder from first division if second division was skipped
    <.[-]                    Output ROT13ed character from copy and clear it
    <-,+                     Read next character
]                            End character reading loop

See also 
 JSFuck – an esoteric JavaScript programming language with a very limited set of characters

Notes

References 

Non-English-based programming languages
Esoteric programming languages
Programming languages created in 1993