If their mother hadn't died.
Once their mother died, Edward and Alphonse decided to do something that was taboo, though they didn't know it: They decided to bring her back to life. Knowing the Law of Equivalent Exchange (which says that you must give something to get something else-like you must study for a test in order to pass), they gathered together all the elements that make up the human body. They prepared a transmutation circle, in which all alchemy had to be done, or else it didn't work. For all they knew. The two young brothers eagerly attempted to bring back their beloved mother, with disastrous results.
Edward lost only a limb. His brother, however, completely disappeared. Edward panicked and somehow managed to bond his brother's soul into a suit of armor. This was at the cost of another limb. Edward lost an arm and a leg. Alphonse lost his entire body.
After this, Edward become a State Alchemist. (A bit like an alchemist for the military.) He and his brother traveled the globe, searching for a Philosopher's Stone in order to bypass the Law of Equivalent Exchange and get their original bodies back. On the way, they discover a plot that could change the world...
I highly recommend this series. However, it's nearly impossible to watch this wonderful anime along with the manga series. If you want to read the manga, which isn't necessary or even a good idea-they don't match at all-you should wait until after finishing the anime. Then you'll have understood the anime more fully.
And you could even watch the second anime!
A good way to watch this would be to pay to watch it on funimation. Or, of course, you could simply look on youtube...