I found this question on how moderators are chosen for public beta and "open site". How were the moderators chosen for the private beta?
There are actually three different types of moderators:
These are employees of Stack Exchange, Inc, such as myself. During the earliest days of a beta, it's easy to tell who the employees are: we're the only ones with diamonds. Community managers are moderators ex officio and remain moderators so long as we're employed by SE.
One of the things we like to see new sites do almost immediately is think about The 7 Essential Meta Questions of Every Beta. Of those, #6 is Who should the moderators be?
Shortly after the public beta starts (1-2 weeks, generally), the Community team chooses 3 pro tem moderators. They're selected (versus elected) based on a number of factors, one of which is community input from a "Who should the moderators be?" meta post—assuming one exists.
Moderators pro tem remain moderators until community moderators are elected.
After a site is launched (that is, has graduated and is no longer in beta) and sufficiently active, SE holds an election. Anyone with a reasonable amount of rep can self-nominate, and the community then votes on who they want to be their moderators.