The Union Church is a place in Sector 3 where players can make "unions". These are proof of the ties of friendship between two players, and nothing more than that. Here is where you can make and accept union proposals, break them, and also see all the union pairings history in the game.
If you are best friends with someone in the game, or even married with another player in real life (that works too!), unions are a great way to show you are great friends with that other person. All that needs to happen is for one person to make the proposal, and the other person to accept. The union pairing will show up in player profiles as a sign of friendship.