So the series of Fullmetal Alchemist is huge, and a good part of the reason why is because of Edward and his brother Alphonse. The writing of these two characters, plus many side characters, and the entire story involved made a phenomenon followed by a good portion of the population who read and love manga.
The back story for Ed is that his father (who disappeared) had a lot of materials laying around, relating to alchemy. Ed and his brother Al found these books and such and started using alchemy. So when their beloved mother passed away, leaving them alone, they went to live with their childhood friend Winry and her grandmother. Not long after, an alchemist came by the town and Ed and Al managed to convince her to instruct them in alchemy. When they came back from her training, they attempted to use what they had learned to transmute their mother back to life. But bringing the dead back to life in alchemy is a taboo. The result was a disaster... Not only was their mother not brought back to life, but Ed's leg was taken and Al had been completely taken into god's realm. Ed used his arm as a sacrifice to bind Al's soul to a suit of armor nearby.
Now, a normal story might end in tragedy right here, but as it so happened, Hoenheim was doing something for the government/military before he disappeared. Ed and Al had written a letter to find out if the military knew where he was. So, after their attempt to transmute their mother, Colonel Mustang showed up in person to respond to their letter. The irony of the situation was that the military strongly prohibited human transmutation (and transmutation of gold). But seeing the strength in Ed, after having a few choice words, he extended and invitation to join the military. So, Ed went through the process of being fitted with automail (artificial metal limbs) for his arm and leg, and rehabilitating himself. A year later, at age 11, he went to Central city for the military alchemist exam. What impressed them wasn't just his ability, but his quick thinking, and being able to manage without a transmutation circle (something that took time and perhaps materials too, to make). When another applicant's transmutation went awry, Ed saved everyone.
At first, Ed is obsessed with trying to find a way to get Al's body and his own limbs back. Its when he finds the answer to how to do it though that the series really starts to heat up. The answer of course being in terms of alchemy.... one life for another. In other words, in order to create the philosopher's stone, which is what they needed, they would need to sacrifice several lives... several hundred... several thousand. So they decided to try and find another way. This is what brought them into something far deeper than they could imagine.
This is when we really start to see Ed start to talk to Mustang and Hughes. Even though Ed insists on insulting Mustang when they meet, often calling him names that perhaps would get normal soldiers disciplined or court marshaled for insubordination, there's an unspoken respect that he has for Mustang. He probably sees that more after he has a talk or two with Mustang's right hand, Lieutenant Hawkeye (a female sniper). The relationship that Mustang has with Ed is that of trust. While Ed doesn't always trust Mustang in the beginning, he trusts his advice.
Later, Ed and Mustang both trust each other completely. This probably starts when Ed finds out about the Homunculus and give Mustang a heads up that the ruler of their country is actually one of the homunculus to. When Mustang sees the proof for himself, and sees what Ed is up against, he decides to fight... for Ed, for the nation, and for himself. Ed at one point borrows a small amount of change from Mustang, and tells him when he's asked to return it, that he'll return it once Mustang is the leader of the country... and then he'll borrow some more change.
Its about here that Ed and Al part ways for a little while. Ed goes up to Briggs, a post commanded by the icy sister of Alex Armstrong. After a tense situation, he manages to convince her to join his side of the fight against the Homunculus and much of Central command. Shortly after, she's shipped away from her post for questioning, regarding the death of one of their men. During that time, a military alchemist (Kimbley) who had been imprisoned previously for murdering thousands in a past war, took over control of the base. By then, Winry had already been sent to Briggs to winterize Ed's automail. Shortly after her arrival, Ed was forced into joining Kimbley to insure Winry's safety. The result was a point where Ed sent Al and Winry (completely against his better judgement) with a past enemy of his, and the enemy's comrades. Shortly after, Ed was impaled by a steal beam of a falling building, courtesy of Kimbley. With some help from his own alchemy and a few past enemies, he somehow made it out alive. From this point on though, he was out of contact with Al and Winry from quite some time, and he teamed up with a rouge homunculus called Greed who had taken over the body of a man who had previously been a friend to Ed.
At this point, the war continues on several fronts against the Fuhrers/Homunculus' plans. One group including Ed and his new companions, another including Al, Scar (Ed's enemy) and his companions, their father (whom they somehow met up with again along the way), and another that includes Mustang and his group that went a-wall from the immoral government.
Though the story itself is completely epic in every proportion, there are times when Ed takes a side trip to get his automail fixed. Now, when he left his town, he burned his own house to the ground. So, he stayed with Winry and her grandmother when he did come home.
His relationship with Winry is rather a minor detail throughout the series, but through the chapters it evolves. At first she appears to be a childhood friend. He jokes with her and barely listens to her, and then he takes her with him to Rush Valley. As they meet again and again throughout, he appears to care more about her. He's not very good at showing it though. But for some reason, he uses Winry's parent's death repeatedly as fuel to propel him forward. Perhaps Winry, her grandmother, and his brother Al were the only people he has ever allowed himself to have any feelings for after he left home. (SPOILER: This is made clear as day when he proposes to Winry at the end of the series.)
Perhaps the most important relationship he has though, is with his brother Al. Al is the force that keeps him from doing some of the stupid things Ed comes up with. Al is also Ed's second guy when it comes to taking out guards, and dodging alchemic explosions of sorts. Al is Ed's partner in a fight, and his conscience when things get bad. He's also his voice of reason on occasion. But another reason why their bond is so close is probably not only because they've always been close as brothers, but Ed used his own blood to bind Al's soul to a suit of armor. Because Al lost his body in the first place, Ed (being the older brother and having laid on the pressure) feels and enormous amount of guilt over the whole thing. And for a long time, Winry and her grandmother are the only ones who know Ed's secret.
As a character, Ed is probably one of the strongest. He has endured such immeasurable pain in his life, that when he confronts it much later, after the fact, it still makes him vomit uncontrollably. That being said, he keeps a slight distance when he meets new people, but this distance is often dashed when Al's warm personality gets involved. But unlike many people would in his situation, Ed doesn't harden himself and turn cold to the world. There are things he doesn't talk about unless he needs to, but he makes no excuses. He also will never pass on helping someone in need, unless its out of his power to do so. Even so, he is extremely stubborn, and often finds ways around the rules to achieve his goals or an ideal that's morally right. Though common sense may elude him from time to time, Ed is probably one of the smartest people in the series. This is even more true when you look at how old he actually is. But with all his knowledge, skill, pride, and ability, he also knows when to let it all go, and he has the strength to go through with it. Out of every manga ever read, this was probably the most successful and the most loved by me, and the writing for Edward and many of the other characters made that possible.