Edward attempted to do a human transmutation to bring his mother back. Durning the transmutation something went wrong. He saw what is called 'the truth'. In return for the information he had to give his leg as a toll for the information. The 'truth' told him almost everything about alchemy, except for the philosophers stone and human transmutation. Using this information, Edward sacrifices his arm so he could bond his younger brother's soul to a suit of armor. (Alphonse's body was taken as a toll for the 'truth') Edward remembered the truth, so he learned how to transmute with out a circle. Alphonse on the other hand lost his memory of the 'truth' so he can not.