Story
First appearance
Fuuma is one of the very first characters to be introduced in the X story, on page 12 of volume 1, in fact. The first we see of him, he's waiting for his sister Kotori to get ready so they can ride to school on his bike. Basic facts about him are established quickly: he is the son of Kyougo Monou, the priest of Tokagushi shrine, and his mother died (quite horribly, in fact) seven years ago.
The neighbours all think that the Monou siblings are examplary children, and Fuuma is the sort of guy who carries groceries for old ladies and saves small children from drowning. As soon as they arrive at school it's made abundantly clear that Crushing On Monou-sempai is quite the widespread affliction among the female population, and that he has earned the respect of his male classmates... not to mention the fact that guys interested in his sister all stay away because they're intimidated by him (*cough* much like the effect one Kinomoto Touya has on his classmates *cough cough*).
In short, Monou Fuuma, at the beginning of X, is the perfect image of a dutiful son, a protective older brother, and a generally all around Good Japanese Young Man. He's good at sports and well-liked, but he's also reserved and stoic.
Awakening
The very day that the story begins, there is a new boy in Kotori's class. As it turns out, it's Kamui Shirou, Fuuma's childhood best friend. Their mothers were close enough so that Fuuma used to call Kamui's mother Aunt Tohru, and Kamui had a bit of a crush on Kotori (he used to say he wanted to be Kotori's wife *ahem!* Moving on...). Kamui made a promise to always protect Kotori, and Fuuma made a promise to protect Kamui as long as he kept his promise.
Kamui and his mother left Tokyo after Saya Monou's very messy death. Fuuma spent the next 7 years basically being a good son and trying to hold his sister together because, in addition to being sweet and docile, Kotori also happened to be sickly and rather on the batty side... understandable, I suppose, given the way her mother died.
When Kamui returns, he's apparently become a rather arrogant little git who won't give his old friends the time of day, but by and by the Monou siblings work themselves back into Kamui's friendship. As it turns out, Kamui was keeping them at a distance because he was involved in messy, apocalypse-type business. He was supposed to choose whether he'd help destroy or save makind, and ends up deciding to save it because he wants to protect "the world where Fuuma and Kotori can be happy".
And that is when all hell breaks loose.
'Kamui'
As it turns out, Fuuma happens to be not just Kamui's childhood friend, but his "twin star", his opposite, born to take the place Kamui will leave vacant when he finally makes his choice. Since Kamui chose to save mankind and become a Dragon of Heaven, Fuuma informs him, it is his destiny to become a Dragon of Earth and kill those who are trying to save mankind. He says the Earth wishes for a change and he will grant it by killing all those who make it impure (i.e. mankind - gotta admit, Greenpeace has got nothing on Fuuma).
To drive his point home, Fuuma impales Kamui to a slab of wall with various glass shards, the Shinken ("Holly Sword") and other other sharp and nasty things, while also licking Kamui's neck and pressing himself up against him; because, in addition to releasing his until-now latent powers, Fuuma evidently also got in touch with his inner seme. He end his performance by killing his sister while Kamui screams impotently. He does this in an impressively messy and through way, involving crucifiction, the Shinken, and Kotori being torn to bits. To be honest, however, Kotori's mind wasn't actually in her body anymore, and the only one that benefited from the performance was Kamui.
Fuuma spends his next appearances filling the role of Big Bad of X to perfection, blowing up buildings, causing earthquakes, and tormenting Kamui. Little by little, however, details are revealed that are evidence that Fuuma is not just a villain bent on destroying mankind. His motives as of now remain unclear; but he has shown other sides of his personality. He can be surprisingly tender with certain people (most often, Kakyou and Nataku), and has shown that he has more humor and charm now than he ever did before.
Most of all, however, Fuuma is uncannily perceptive, to the point that he can sense people's deepest desires. On several occasions, he has granted said desires. However, the way in which he grants this desires is reminiscent of stories about genies that grant their "master"'s wish in unexpected and often lethal ways. People often write this off as sadism on his part, but the later volumes of the manga shed a different light on Fuuma's wish granting. Fuuma does not do what he does because he gets off on twisting people's wishes, he does it because it is trully what the person desires. It's a running motif of X that people should pursue their true wishes, whatever will make them trully happy, even if society condemns said wishes as wrong. Fuuma is merely enabling people to reach what they want; as he says it, he expresses his own wish whenever he grants anyone else's.
Speaking of which, the biggest mystery surrounding Fuuma is the matter of his own wish. In one of the last published installments of the manga, it was been revealed that Fuuma's wish can only be granted by Kamui, who remains oblivious not just to Fuuma's but to his own true wish. It's safe to say that the Earth's fate rests on both of those wishes.