A private union is created by a real player, but not by the server. You can become a member of one of these unions only after the union leader approves your request. Sending a request is easy, all you have to do is press the button at the bottom of the screen.

You can create your own private union as well. Read the corresponding article about it.
You can send only one request at a time. While your request is active, you cannot join other unions. If you want to cancel your request, open the union window and press the Cancel Request button.
Keep in mind that when you change unions, all the points you have gained will stay with the previous union and will not transfer with you to the new one. Also, if you forgot to collect the reward for a completed quest before leaving, it too will stay in the previous union.