Making Friends the Easy Way:
If you click or tap on another player's avatar a "mini profile" opens with a number of options. One of those options is "Add Friend". Select that and that player will get a pop-up asking if they accept your request. If they agree they're added to your friends list. If not, nothing happens. Some players have deactivated this feature, but you may still ask them to be friends the not-quite-as-easy-way.
Click the Add Friend button in a player's mini-profile. If they accept, you'll each be added to the other's friends list instantly.
Making Friends the not-quite-as-Easy Way:
You can also invite somebody to be your friend from their main profile page. You will find the Invite Friend button in the upper left-ish part that screen. If you ask them that way, they get an Email message asking if they accept. This is how you can ask somebody who isn't in the room with you to be your friend.
You can also remove a friend by re-selecting the friend button from that player's Profile.