Fate Core is a generic implementation of the fate engine, designed as a manual for both gaming groups and developers to design their own settings and campaigns. It is available for free (see later in this wiki), and licensed under creative commons or the OGL.
The Fate Accelerated Edition is a simplified version of the Core, suitable for shorter or high-concept campaigns and introducing new players to the Fate engine.
Both Core and fate-accelerated mechanics are compatible with each other and other Fate-based games, at each group or developer's discretion.
Those interested in tinkering with the Fate Core rules may be interested in the Fate System Toolkit, which is a compilation of rules modifications and customisations, some of which have seen use in further adaptions of the Fate system such as Atomic Robo RPG.
How can I get the books for Fate Core, Accelerated, and the System Toolkit?
There's three ways to get Fate Core, and two of them are free (PDF or online SRD). The free versions are full versions of the rules, not cut down or minified.
PDFs: Evil Hat offers the PDFs for Fate Core on their website as a free download, on a pay what you want (including free) basis*. Fate Accelerated and the Fate System Toolkit can be found on the same page.
These PDFs contain the same content as the physical books, including illustrations and everything. For those wanting the books, you might want to check out the PDFs.
Online Fate SRD: You can use the Fate SRD, which provides a complete online reference for Fate Core, as well as Fate Accelerated and the Fate System Toolkit. This is pretty handy for linking to, or looking things up on the spot.
Books: You can buy the books from Evil Hat's Store, your friendly local gaming store, or other distributors. Here's the links to the Evil Hat store:
All three books are available for free under the creative commons license or OGL, which is why the Fate SRD can exist.
* Those curious about why they provide it as pay what you want, including free, may be interested in reading Fred Hicks's blog post about Pay What You Want. For reference, Fred Hicks runs Evil Hat.