One of my favorite things about Persona 3 is the character Fuuka and her role as a kind of distant information analyst and navigator for the party. Even ignoring Fuuka's role as a character in the story, she serves an incredibly valuable role in the exploration of Tartarus, and her presence is one of the things that makes Persona 3 so unique. One of the most important things to mention about her is the fact that, even though she never participates in battle directly, she gains experience, grows in level, and learns new skills just like any combat character. A lot of her individual abilities are very similar to the kinds of things you see in other games, but the fact that they are tied to a particular character is what makes Fuuka unique.
First off, Fuuka's most important role is battle support. She does not participate in battle directly, but she observes battles from a distance and can be ordered to perform actions through the same Tactics menu that commands every other character. Her main battle command is Full Analysis, an upgraded version of the Analysis command that the original support character, Mitsuru, uses. Mitsuru's Analysis only identifies an enemies Arcana and weaknesses, but Fuuka's Full Analysis reveals all of that and the attacks an enemy can use. Also, as Fuuka increases in level she eventually learns Oracle, an ability that will trigger a random effect once per battle when called upon. Both of these abilities only take effect when called upon, but they are both controlled by the support character's own abilities and are independent of the composition of the battle team. Most RPGs that have such effects link them to a battle character (so you can only use something like Analysis if you bring a combat character who has that ability) and force you to use up a battle characters turn in order to use such an ability, but Persona 3 doesn't, so abilities like Analysis are reliably available and much more easily used. In many ways, this set-up makes many kinds of abilities that could otherwise be a poor choice into something incredibly useful.
In addition to her active combat support abilities, Fuuka has a few passive ones as well. Fuuka's Support Scan and Third Eye abilities both change the information available to the player in normal battle. One lets you see what buffs and debuffs affect selected targets, and the other modifies the main target selection cursor so it will tell you whether an attack will be effective or not. Both of these abilities are unlocked as Fuuka rises in level, so it can be said that raising Fuuka's level changes the game interface and the kind of information available to the player.
Finally, Fuuka has several important roles and abilities relating to the exploration of Tartarus. She provides you with information about what is happening, such as telling you about unusual events that occur on some floors, warning you when a character is low on health, or warning you when the Reaper appears on a floor. This means that important gameplay information is relayed to the player through a character in the game, which is always a good thing. As Fuuka raises level, she gains the ability Healing Wave, which heals the whole party whenever you climb a set of stairs, and Escape Route, which lets you warp back to the entrance whenever you like (making Access Points and Traesto Gems obsolete). Much like some of her combat abilities, these abilities essentially change some of the rules of the game, and tie those changes to Fuuka's level growth. Simply put, it becomes much easier to climb Tartarus as Fuuka rises in level. Overall, linking rule changes to a character's growth is a very elegant way of allowing the player to have more options as the game progresses.
My only complaint about Fuuka's role in the game is that they could have given her so many more good abilities. Perhaps she could have gained abilities that improved the game's mini-map/radar, an ability to read how much longer you can last before your characters grow tired, or an ability that made finding rare treasures easier. Certainly they could have given her more combat abilities, such as the ability to read what attack an enemy will use next, or some other direct combat effect like Oracle. Other than that, though, I think she does nothing but contribute greatly to the game.