"Alchemy is the reconstruction of matter in new forms based on the knowledge of natural laws. This world flows by obeying these laws. Death, too, is part of that flow. Accept the flow of the world."
Traditionally alchemy is the quest to change 'worthless' base metals into gold via magic or means of chemistry.
Alchemy is the ability to change an items form and shape at will with a sort of "medium" called a transmutation circle. Transmutation circles are generally complex circle shaped markings, and the type of transmutation circle depends on the type of transmutation involved. Say, if the transmutation requires that someone simply change a log to a wooden horse, you would need a certain type of circle for it to be effective.
However, there are still multiple ways in which some people may bypass these laws, such as Ed and Izumi Curtis (the Elric brother's mentor and a respectable alchemist herself, though she doesn't like to use it to make her life simpler) seeing the Gate of Truth in order to become as skillful in the art as they are.
Though at first the power of Alchemy may seem unlimited, there are certain physical laws that it cannot bypass by any normal means. To transmute something into something else, it must have the same properties (i.e. the log and wooden horse) and equally importantly they need to be of the same mass. It is against the laws to transmute a log into an entire cabin, for example.
The Alchemical transmutations are all based off of the four elements fire, wind, water, earth, and the circles they are placed in. All of these things heavily affect the outcome of the item. In example, say a certain circle can change a piece of iron into a hammer, and another would turn it into a spear.
There are a certain exceptions to the entire necessity of a transmutation circle as a whole. First off, there is Edward Elric, the Fullmetal alchemist. He can use Alchemy without use of a transmutation circle because he has done the unthinkable by attempting to bring back human life. Because of this, Edward Elric was sent to "The Gate" and saw "Truth", a strange malevolent being that took what he wanted when he wanted. Also, Izumi Curtis, Edward Elric's teacher, has use of this along with his brother Alphonse, who has also seen "Truth".
Along with seeing "Truth", there is also another way to use Alchemy in an instant way: the use of transmutation Circles on your arms or hands at all times. This is what Colonel Roy Mustang does to shoot the flames out of his gloves through the air and friction, a very strategic move for combat.
There is also one other way to gain power of Alchemy and bypass the laws, and that was by using the legendary artifact: The Philosopher's stone. This legendary artifact is very elusive and it seems that no one has been able to find it...it is wondered if it even exists.
There are many Alchemists in the Fullmetal Alchemist world that can perform Alchemy, also known as the devil's magic. Many people, like the Ishbalans, believe that it is an unholy act against the Gods to change an objects form or shape, like Scar, the Ishbalan serial killer who is currently targeting the State Alchemists all across Central due to his holy beliefs of the God of Ishbala.
Edward Elric- he is able to use Alchemy without a transmutation circle, as he saw "Truth" when he went through the gate after he tried to bring his mother Back to life...at the expense of his left leg. He would later lose his arm by trying to save his brother Alphonse Elric, as rather than him losing a limb like Ed did, Alphonse lost his whole body.
Roy-uses the transmutation circles on his gloves and the friction from when he snaps his fingers to create fire in dry air. He is all but useless in the rain.
Scar - his right arm is not his, it was his older brothers, who made his entire body a transmutation circle before the entire Ishbalan race was destroyed. Scar attempted to fight one of the (insane) State Alchemists as they ran from the attackers, and with two swipes of his finger his forehead basically exploded into the scar we see today. He would later lose his arm to the same State Alchemists, and his brother , knowing death was only a few hours away, Scar's brother took off his own arm and attached it to Scar with Alchemy. This is one reason that Scar hates alchemy, but uses it, as his brother died from blood loss shortly after.
Alphonse was originally just like any other alchemist, albeit slightly better due to the fact that alchemy kept him alive. However, after seeing one of his closest friends die, he was able to understand the Gate of Truth, the very same one which Ed saw when he tried to bring their mother back to life. This now allows for Al to use Alchemy without a circle, just like Izumi Curtis and Ed.
Equivalent Exchange was the idea from the anime of the Fullmetal Alchemist series that most alchemists referred to when explaining the laws of alchemy and in many forms of trade in real life, always saying that in order for them to do something they need the right materials. In essence, they meant that a river couldn't come from a raindrop, at least. not from them.
They would need the appropriate amount of a certain element before attempting a transmutation at all, if not, the transmutation would simply cancel out, or, in much more serious attempts, the effects would be much, much more drastic, such as losing organs like Ed and Al and Izumi did in their try at bringing a human being back to life. This resulted in a lot of things happening, but none of them being positive to the slightest extent.
In the anime, an attempt like this simply made a homologous, but it has nothing to do with that as far as we know in the manga series, and doesn't even seem to really make an indifference, as the things that people have always brought back, as far as Ed and Al know, have only been random human parts, only barely even resembling the original person at all. In fact, Ed was put through much hardship when he discovered that the thing that he brought back from the dead was not his mother at all, meaning that the equivalent exchange didn't exist in that trade at all, giving him nothing and taking nearly everything.