The "Private Beta" period of a new Stack Exchange site is a limited time period during which only users who have expressed interest in the site's proposal can participate.
When a new Stack Exchange site is proposed on Area 51, it goes through 3 phases: Definition, Commitment, and Beta.
During the Commitment phase, users can commit to the new site. This means they agree (commit) to make at least 10 postings during the private beta period.
If enough users commit, the site enters private beta. The committed users then try to post a sizeable number of questions and answers to ensure that the site gets into public beta. Only users who have either originally committed or have signed up for the site through the Area 51 proposal page are allowed to access the site.
During private beta, the reputation requirements are extremely low to allow users to quickly gain moderation abilities in order to help the site through its first few weeks of beta. A site typically stays in private beta for about 2-3 weeks before it can move on to public beta.
If the private beta is successful (generally gauged by at least 150 high-level questions), the site will then move on to public beta. As the name implies, during public beta everybody can sign up and participate.
For more information, see: