After a proposal from Area51 (such as the Emacs StackExchange) is accepted, it enters a private beta phase for 1 to 2 weeks. During the private beta, the site is semi-closed to the public. A core group of users who committed to the proposal or joined shortly after the private beta started work to give the site some momentum and establish norms, tags, etc. This site first "launched," entering private beta, on September 23, 2014.
Once the Team (Stack Exchange developers) feels that the private beta has been a "success" and that the beta site has the potential to become a thriving community of its own, the site enters a public beta. For all practical purposes, the public beta is the *official* launch of the site. This is what the Area51 FAQ says:
Beta. Perhaps the most important phase. This is the actual, live site set up on a "probationary" basis to see if people use it. It is very important to participate early. The earliest questions set the tone and topic of the site for a long time. This is also the time to spread the word via Twitter, blogs, and email far and wide. If the site does not get used, it will be deleted. Each site has two parts, each with its own URL:
topic-name.stackexchange.com: This is where you ask questions, answer questions, tag questions, edit questions, and vote. The beta site starts off with a temporary "sketchy" design; its final design is chosen when the beta period ends.
meta.topic-name.stackexchange.com: We don't want to talk about the site on the site itself, so each site has a meta discussion area to talk about things like what questions are appropriate, what tags to use, etc. The meta site is also accessible through the 'meta' link at the top of each page on the main site. This is where you can help make decisions about the site itself.
As of October 7, 2014 the Emacs StackExchange is officially in Public Beta. Admittedly, this came with very little fanfare. I at least expected to see a post here in Meta about it. So, for those waiting for the Meta post like me, this may be the best you get.