Programming in the early days of computing was tedious in the extreme. Programmers required a detailed knowledge of the instructions, registers and other aspects of the central processing unit (CPU) of the computer for which they were writing code. The source code itself was written in a numeric notation, so-called octal code. In the course of time mnemonic codes were introduced, a form of coding known as machine or assembly code. These codes were translated into the instruction words by programs known as assemblers. In the 1950s it became increasingly apparent that this form of programming was highly inconvenient, although it did enable the CPU to be used in a very efficient way.
The first FORTRAN compiler was a milestone in the history of computing, at that time computers had very small memories (of the order of 15KB, it was common then to count memory capacities in bits), they were slow and had very primitive operating systems, if they had them at all. In those days it seemed that the only practical way is to program in assembly language.
The pioneers of FORTRAN didn't invent the idea of writing programs in a High Level Language (HLL) and compiling the source code to object code with an optimizing compiler, but they produced the first successful HLL. They designed an HLL that is still widely used, and an optimizing compiler that produced very efficient code, in fact the FORTRAN I compiler held the record for optimizing code for 20 years!
This first FORTRAN compiler was designed and written from scratch in 1954-57 by an IBM team lead by John W. Backus and staffed with super-programmers like Sheldon F. Best, Harlan Herrick, Peter Sheridan, Roy Nutt, Robert Nelson, Irving Ziller, Richard Goldberg, Lois Haibt and David Sayre. By the way, Backus was also system co-designer of the computer that ran the first compiler, the IBM 704.
The earliest record I have seen a reference to is "Preliminary Report: Specification for the IBM FORmula TRANslation System, FORTRAN" published as an IBM Applied Science Division report on 10 November 1954. It is not in the title, but the design was explicitly for the IBM 704.
The programmer's reference manual "The FORTRAN Automatic Coding System for the IBM 704 EDPM", IBM doc 32-7026, was issued in October 1956 and a primer, "Programmer's Primer for the FORTRAN Automatic Coding System for the IBM 704", IBM doc 32-0306-1, appeared in 1957. Software for the IBM 704 was released "early in 1957".
IBM itself seem to regard 1957 as the official "birthdate" as they produced a film "The Origins of Fortran" in 1982, the 25th anniversary.
The new invention caught quickly and no wonder, programs computing nuclear power reactor parameters took now hours instead of weeks to write, and required much less programming skill. Another great advantage of the new invention was that programs now became portable. Fortran was adopted by the scientific and military communities and used extensively in the US Space Program and military projects.
The phenomenal success of the FORTRAN I team, can be attributed in part to the friendly non-authoritative group climate. Another factor may be that IBM management had the sense to shelter and protect the group, even though the project took much more time than was first anticipated, the original 6 months timetable stretched to 3 years!