The Free Software Definition is stallman's formal specification of what makes software "free" in the sense he intends. Published on gnu.org (first canonically in 1996, though the underlying ideas are older), it states that a program is free software if it provides all four-freedoms to every user: the freedom to run, to study, to redistribute, and to modify and redistribute modified versions.
The definition exists to settle a precise question: not whether software is free of charge, but whether it respects the user's freedom. The libre-vs-gratis and free-as-in-freedom-concept distinctions exist precisely because English conflates these two meanings of "free."
Structure of the Definition
The Free Software Definition is binary: software either satisfies all four freedoms or it does not. Partial satisfaction — software that permits three of the four — is still non-free software by Stallman's definition. This all-or-nothing structure reflects his philosophical position that freedom is not divisible: a single restriction is enough to subject users to the developer's power.
The definition specifies freedoms, not features. It says nothing about price, quality, documentation, or developer community. Software can be expensive and still be free software (in the relevant sense); software can be gratis and still be non-free if it withholds source code or restricts modification.
What the Definition Excludes
The definition is explicit about what does not make software non-free: requiring attribution, requiring that modifications be released (copyleft), requiring that the original author's name not be used to endorse derivatives, or requiring that derived works use a different name. These are conditions on distribution, not restrictions on freedom.
What does make software non-free: restricting the right to run (use-case restrictions), withholding source code, prohibiting redistribution, or prohibiting modification. Any one of these disqualifies a program.
Relationship to Open Source
When eric-raymond and bruce-perens founded the open-source-initiative in 1998, they produced the Open Source Definition — a parallel document with ten criteria for open source licenses. The OSI definition is substantially similar to the Free Software Definition but admits some licenses that stallman considers non-free (particularly licenses with field-of-use restrictions or discriminatory clauses). stallman's position is that the two definitions mostly agree in practice but differ in philosophy and occasionally in specifics — see software-freedom-vs-open-source.
Living Document
The Free Software Definition on gnu.org has been updated over the years to address new edge cases — tivoization, SaaS, DRM. When the gpl-v3 was drafted, the definition was revised to clarify that Freedoms 1 and 3 require that modification and installation be practically possible, not just legally permitted. This is the philosophical underpinning of the anti-tivoization clause in GPLv3.
Authority
The Free Software Definition is maintained by the free-software-foundation and is the authoritative statement of what counts as free software for FSF purposes. The FSF licenses list and the GNU package acceptance criteria both use it as the test. It is the foundation on which stallman's entire project rests: every argument he makes about proprietary software, every campaign against DRM or software patents, presupposes this definition of what freedom means in the software context.