Today, with the release of Firefox 43, almost a decade after the idea was first mooted, there is finally an official 64-bit build of Firefox for Windows. To download it, you'll have to head over to the Firefox website and explicitly grab a 64-bit installer; if you just do an in-place upgrade you'll just get the normal 32-bit flavour.
64-bit Firefox for Windows is mostly identical to 32-bit Firefox for Windows, except that very few plug-ins will work with 64-bit Firefox. This is by design: Mozilla is in the process of dropping Firefox's support for NPAPI plug-ins. NPAPI support is being dropped due to (ostensible) stability and security concerns. Amusingly (or ironically), though, 64-bit Firefox does still support one plug-in: Flash. Sites that use other NPAPI plug-ins, such as Silverlight or Java, are being told by Mozilla to "accelerate their transition to Web technologies."
Over the years there have been a number of unofficial and alpha/beta builds of 64-bit Firefox for Windows, but they've always been aborted before they made it to the stable release channel. Back in 2012 an executive decision was made to halt 64-bit builds entirely due to "significant negative feedback" and a frustrating user and tester experience—but a few months later that decision was reversed and the 64-bit builds continued, albeit very quietly.
In 2014 Mozilla confirmed that a stable build of 64-bit Firefox for Windows was finally on its way, with a planned release of "before the end of 2015." It looked like Mozilla was on target for a 64-bit version of Firefox 41, which was released in September, but alas it wasn't to be. Today, though, with the release of Firefox 43, it has actually happened.
Well done Mozilla: that's one big headache finally out of the way. Next up: Shipping Electrolysis (e10s), the long-overdue update to desktop Firefox that will allow it to use multiple threads/processes, like Chrome.