A fanlisting is simply an online list of fans of a subject (in this case for Rhys) that is created by an individual and open for fans from around the world to join.
There are no costs, and the only requirements to join a fanlisting are your name and country. Fanlistings do not have to be large sites (although some are), they are just a place where you can sign up with other fans. TheFanlistings.org is the original (but not official) web directory for fanlistings, dedicated to uniting the fans.
Description from TFL.

Rhys is a healer, a mercenary of the group lead by Grail and later by Ike. He's the first healer you acquire in the game (and also the best one) and the only light magic user you can get in all Fire Emblem Souen no Kiseki.
Rhys is a shy person, he doesn't like to be involved in fights and he always worries for the safety of his companions. The name of the site comes not only from this particular skill (Equilibrium) that is associated to him, but also since he's the most equilibrated person in your party. He's somehow frail and he gets ill really quickly, but he doesn't complain about his physical weaknesses. He's particullary caring about Ike and Tiamat (of course they're the best couple in the game!).
We also know he lived in a small village and he's son of farmers (probably) and also he wasn't very healthy. He chosed to join the mercenaries to earn some money to sustain his family and to show the others (but mainly to himself) he's capable of doing something useful for others.