The Buddy feature has been here since the start of the game. It allows players to add each other as in-game contacts so they can easily meet and visit each other's houses. The feature has changed slightly over the years.
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 you appear in each others 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 relationship request to another player, you must be buddies.