FRIENDS: "Friends" are Facebook friends that also play the game. Anyone that you are friends with who plays the game should be automatically added to your friend bar at the bottom of the game screen. To make friends, simply post your link in the game chat, or many of the FB Groups also have an "add me" thread to which you can add your name and ask for people to send them a friend request. Once you have friended someone via Facebook, simply refresh your game and the person should be in your friend bar.
Many people don't want strangers seeing their Facebook, and that's completely understandable. You can put game friends (or anyone, actually) into the "Restricted" category when you friend them, and then they will not be able to see anything that you post unless you have that post set to Public. You can also "unfollow" the person and then you will not see their posts in your own newsfeed.
To restrict someone, go to their profile and hover over the "Friend" button on the right, and a drop down menu will appear. Choose "add to another list" and click that. Scroll down through your lists until you come to Restricted and choose that. If you like, you can also set up a list for all game friends and also add the person to the list. This helps you find all your game friends for a particular game easily. To unfollow a person, hover over the "following" button and in the drop down menu, choose "unfollow".
NEIGHBORS: "Neighbors" are found only within the game, and have no access to your Facebook account. This is good for mobile players and for those who dislike adding people to their FB account. To add neighbors, go to the end of the friend bar and click on the "Neighbors" button.
When you click the neighbor button and it opens, the middle tab will be automatically selected. You should have a bank of 4 pictures of players, with the words "send a request" next to each one. The words are not brightly lit, but they are clickable. Click the "send a request" next to each person you want to send a request to. If you don't want to send requests to anyone on the page, click the refresh button for a new set of players.
Clicking the third tab at the top left takes you to the area where you can cancel requests that you have previously sent out. You can only have a certain number of active requests, so if you want more neighbors, you have to periodically clear out the requests left on the third tab.
The first tab of the window takes you to list of neighbors who have accepted your requests, and also the players who have sent requests to you - you can accept those requests here. If you ever decide to remove any neighbors, you can do it on this first tab, just click the red X next to their picture.
Neighbors will also show up in your friend bar at the bottom of the game, but since you are not Facebook friends with them, there is no way for you to contact them or for them to contact you.
You cannot search for someone you know in the neighbor section. Occasionally, but VERY rarely, you may actually see a player you know, but the selection is totally random.