Game center is an independent app from APPLE. every iPad and iPhone has this app. Many of the games you play on your phone or iPad use game center to add another level of achievements or community to them. You can be friends on Game Center without playing hay day. But every friend you have on game center can be your Hay Day friend if they play the game.
I had maybe three Game Center friends (my two small kids and a neighbor boy) We all played different games. Two of them used to play hay day but don't anymore. All of my other Game Center friends have come from this game so you can see how well Game center has integrated itself into this game.
To get more Game Center friends you need to know their Game Center ID or make use of Game Center's recommendation feature. With permission, it can go through your contacts and recommend any of them who have set up a game center ID. It can also see what Game center connected games you play and recommend people who play the same game or who share friends with you. Those might not be forum members. If you want to add folks who have used this forum, look either in the special thread here for folks also looking for more friends or look to see if the person you are interested in has included their GC Id in their signature like I have and like MarySC has and others. If they freely give their GC id, that means they don't mind having new friends. Copy that GC id and add it to game center like Mary explained above. You will get a notice if and when that person accepts your request. Likewise you will get a notification when someone has added your GC id and put in a request to have you added as their friend. You can accept or ignore this request.
You don't contact or play with anyone THROUGH the Game center App. They are added to your list of friends in any game that you both have in common. For instance I used to play Tiny tower. You didn't need friends to interact with in that game, but if you wanted to see their tower you could do that. I only had my own children as friends then so I could only see two other towers. Now all of my hay day friends who also played this game have towers i can now see. I have played a lot of games but this is the only one that I felt the need to add Game center friends in
I hope that explains Game center a bit more for you. It is essentially a harmless app that coordinates with other games to allow a social aspect to these types of games and to connect players with games
