What is a "private beta"? Is it related to the Beta badge? How does one get the Beta badge?
See the question Should the beta badges stop being awarded to users? about awarding of the Beta Badge.
You can also check out the FAQ on Area 51 to learn about the Beta Phase:
When a Stack Exchange site goes from Commitment Phase to Beta Phase, for the first week, only those users who Committed to the proposed site will have access; this is so they can prime the system with content before it goes public.
Back in the day, Stack Overflow, Server Fault (and all those other links at the bottom of this page) went through their own private Betas; all users who participated in them (back in the day) got the badge.
So that boat has kind of sailed; you can't get a Beta badge on SO, Meta.SO, SF or SU.
But you can commit to proposals on Area 51 and then get the badge on those sites.
In the context of Stack Exchange? A period of semi-public testing open only to those who "committed" to the site. The private beta will be followed by a "public beta" open to all, which may be followed by graduating the site to fully active status (or may not, but we don't know what the graduation conditions are).
In the context of Stack Overflow, Super User and Server Fault, is was a semi-public testing period open by (easily obtained) invitation. The Stack Overflow, Sever Fault and Super User private betas are long over and if you didn't participate you can't get the badge.
::invokes stern self control to not stick tongue out at late comers::