"It's in the genes that girls will like fortune-telling."
Fujibayashi Ryou is a somewhat-often-present minor character who has some extra prominence during roughly Episodes 15-18 of Clannad. She doesn't have a full arc for herself: most of her significance derives from her attachment to her sister.
Ryou is a fairly reserved high school senior with a strong bond to her sister and a passion for fortune-telling; she's also a class representative, like her sister. She tries to assert her authority as a class representative, but she's too timid to be effective at it. In particular, she tries to get Tomoya to come to school on time, which never succeeds. She knows the ins and outs of the school, but she doesn't seem to have many good friends (probably due to her timidity). Her best friend is her older twin sister, Kyou, who is excessively and violently protective of her. This makes some sense, since it seems Ryou is easily made uncomfortable and possibly scandalized by things (like the "bi" incident with her sister). Her biggest quirk is her passion for fortune-telling: she loves it in all forms, but it seems she prefers to use cards herself. The fortune-telling isn't meant to be totally serious, though: she knows there isn't just one predetermined future, there are many possibilities. Due to this, her fortune-telling is more for fun and games than anything else. Her first appearance showcases her quiet, unassertive personality and her fortune-telling passion well:
Ryou's story is largely tied into her sister's story, because they are usually together, and her sister is a much stronger personality than she is. As a class representative (and one who takes her job more seriously than her sister, it seems), she helps to get the drama club reinstated, even with all the student council's directives and requirements. She's first introduced into this by some tricky on Sunohara and Tomoya's part, resulting in a very humorous scene early in the show. They tell her someone is going to confess his love to her, and it turns out to be Nagisa asking for help with the drama club. She agrees to help as best she can. This involves explaining how many members to acquire and how to get members, along with the requirement to have an advisor. Basically, she is like the liaison between the drama club and the student council.
Besides this background role of helping the drama club (and being, at least nominally, a member as well), she comes into more prominence after Nagisa falls sick and must stay at home for a few days. Ryou has always had some feelings for Tomoya, and her sister decides now is the perfect time to act on these. Kyou sets up many lunch dates for Ryou and Tomoya, and she even takes them to use a love fortune-telling machine (which backfires). This last incident does result in a conversation between Tomoya and Ryou alone (although it doesn't amount to anything romantically).
With all the prodding her sister does to get her and Tomoya together, Ryou's attempts in the end fail. After the tennis injury where Tomoya vigorously defends Nagisa, Ryou is in tears like her sister (although this may be more out of apology to her sister than for losing Tomoya: see Themes section below).
Following this incident, she becomes a quiet background character again, being involved in the group of friends, but never really having any defining moments. She's present most times the group is together (and most times her sister is anywhere). After high school, she heads off to nursing school. At the New Year's Party, she comments on how life is mysterious, and she reiterates her belief that there is no one predetermined future: there are many possibilities. During the final episode montage, we see her, now a nurse, working in the hospital, smiling to brighten her patients' day.
Effect on Main Plot
Like her sister, there's not really anything supernatural in Ryou's story. Her fortune-telling, though, could be considered in that realm: if it worked, it would definitely be supernatural. Her acknowledgement of life's mysteriousness could be seen as an implicit recognition of the supernatural, and her statement about the fact that there is no predetermined future is definitely an important idea for understanding the ending.
Besides that, her main help to the plot is her working for the drama club. Without her knowledge of the student council and her helpful spirit, the drama club may have never made it to the point of performing Nagisa's play. Considering that this drama club is a big factor in bringing Tomoya and Nagisa together, she could be seen as playing a role in that, by extension. She's also another girl Tomoya passes by as he realizes his devotion to Nagisa.
Thank you for reading. God Bless, and peace.
Nota Bene: All clips are from the Clannad Central YouTube channel run by the Clannad (クラナド/Kuranado) fan page on Facebook. All character themes and other music from the show can also be found on said fan page, in the music player. My gratitude to them and all the work they do.