You can join a Fellowship by invitation or by sending an application. In order to apply for a Fellowship, the best method is to open the Fellowships window, go to the 'Requests' tab and click the 'Find Fellowships' button. The game will give you a selection of Fellowships that you can join. If the results don't fit for you, you can ask the system to find a new selection for you. Alternatively, you can view the rankings underneath your player avatar, from there, you can select "Fellowships" and apply to one that suits your game-play. You can also accept invitations from Fellowships by clicking on the shield icon next to your player avatar and clicking the appropriate boxes to accept or decline an invitation.
Some Fellowships allow you to join them immediately, without the need of any application process. Such Fellowships will have a 'Join' button, rather than an 'Apply' one.
Alternatively you can of course found your very own Fellowship and invite other players to join you.
If you have joined and left a Fellowship recently, you will not be able to join another for 24 hours.