When Edward and his brother Alphonse were just boys, their mother succumbed to a sudden illness. Seeing no other alternative, the two of them attempted to revive their dead mother using what they knew: the science of alchemy. However, alchemy still obeys the principles of nature. The most basic principle of alchemy is that of Equivalent Exchange: for everything, there is a price. You cannot create something from nothing.
Edward and Alphonse knew of Equivalent Exchange and thought they had everything figured out. What they learned--disastrously--is that some things just can't be measured. Edward gave of his own body in the attempt, losing his left leg up to the thigh. His right arm was lost in a desperate move to prevent the soul of his brother (who lost his whole body in the attempt) from being lost forever.
After the incident, Edward was fitted with an automail arm and leg (automail is a special bio-mechanical prosthesis that can move at the wearer's will, enabling virtually full function). Recovering from his failure, Edward pushes forward with his pursuit of alchemy with a new goal: undoing the damage wrought upon himself and his brother. Just about a year after the accident, at the age of 12, Edward became the youngest certified State Alchemist in history. Though Edward is reluctant to join the State Alchemists (which is essentially part of the Army), he becomes one so he can research the Central archives for a possible solution: the Philosopher's Stone. The stone is the holy grail of alchemy. With it, it is said, one's power over alchemy rises to divine levels, allowing one to bypass the limitations of alchemy, including Equivalent Exchange. It's Edward's wish to find the stone and use it to restore his limbs and Alphonse's body.
Obviously, Edward is a top-notch alchemist now at the age of 15. Because he wears automail, he was given the title "Fullmetal Alchemist" by Fuhrer King Bradley when he became a State Alchemist (all State Alchemists are given such titles). Besides his incredible skill for his age and proficiency with weapons, Edward has one other secret that shows his genius. He is able to perform alchemy without the help of a Transmutation Circle or Array: normally a requirement (it was in fact that feat that made him worthy of the State Alchemists). As a result, he has amassed a reputation throughout the land, and this notoriety becomes a potential liability because people start gunning for him: not necessarily because of his identity but also because he's a State Alchemist (or "Military Dog" as some state-haters would call him). Edward answers to Lt. Col. Roy Mustang, and in exchange for answering to him and following his orders, he searches for the Philosopher's Stone in between his assignments and reports on his findings.
Because he's always in the company of his brother Alphonse (who is now just a suit of armor), people sometimes mistake Alphonse for the Fullmetal Alchemist. The problem is compounded by the fact that Edward is short for his age, so some people think he's younger than Alphonse when he's actually a year older, which REALLY pisses him off and causes him to flip out whenever someone points that out.
In addition to his alchemy skills, Edward can also kick serious butt. His tough automail arm and leg can deal some serious damage and can provide nice protection from physical attack. In addition, Edward can transmute his prosthetic arm into any of an assortment of weapons to suit the occasion.
If there's a weakness in Edward, it's in his naivete. Being so young, he still hasn't seen the depths of human depravity, and each new dive into the dark side of humanity frightens him. In time, he develops a sense of sarcasm to try to fight the fear.
Apparently, in Japan the term "Full Metal" means a very stuborn person... which describes Ed exactly!