You have to be FaceBook friends before you can ask someone to be your neighbor in FarmTown. To access your neighbors list, click on the "My Neighbours" tab at the top of your FT page.
The Neighbors screen shows all your friends who play Farm Town and shows whether you are  neighbors.  If you wish to add someone who isn't, simply click  Add as a Neighbour. You can remove neighbors you wish to replace or who no longer play by clicking on Remove Neighbour. You have Sort options making it easier to find what/who you are looking for; whether it's their Activity, Level, Diversification, or Help Given. Sorting by Name puts them in alphabetical order. There is also the option to show only Active neighbors; you can even send a gift from this page.  To visit your friend's farm--whether neighbor or not--click on their pic or name.
At the bottom of My Neighbours page you have the opportunity to invite your friends who don't play Farm Town. The MY NEIGHBOR'S TAB shows all your neighbors plus any friend who has played FarmTown. You can scroll down the page to find the friends you want to Invite, or you can type their names into the search box.
When adding a neighbor, if there is not enough space available, you will get a notification. To add more neighbors, you will have to delete some you already have.  Look for ones who have not been playing FarmTown recently.