The first step in adding a player to your Buddylist. Buddy Requests are made from the Galaxy view screen. Enter the Coordinates for a planet controlled by the player that you want to add. In the action column there is an icon of a green sphere, this is the Buddy Request icon. This opens the Buddy Request dialogue, where you can type your request to add that player to your list. If that player wishes to add you to their buddylist, they must also send you a request. Requests can be retracted prior to being accepted by clicking the Own Requests link on the Buddylist page.
When sending a buddy request you should probably send a message with it saying that you have sent a buddy request, as the player has to go look at the requests section to see the request. There is no obvious notification.
Please be forewarned that anyone you accept as your buddy will always know when you are on-line; seems kind of strange to think of a "buddy" in this way but just because someone wants to be your buddy doesn't mean they aren't out to get you. Being your "buddy" does not prevent them from attacking you. So requesting that someone become your buddy probably isn't a good idea if you don't trust them. Only accept/ask for a buddy if you trust them.