How it works
To add somebody as a buddy, you click on their avatar and at the bottom of the menu it will say "Add as a Buddy". Click on this and a request will be sent to the other person. If they accept, you will become buddies and they will appear in your buddy list, and likewise you'll appear in their buddy list. From the buddy list you can see if they are online and if they are you can join them in the game. If they are offline you can go to their apartment and leave a message.
In order to send a Relationships request to another player, you must be buddies.