The transmutation circles contain the formulae and/or specific incantation needed to carry out a function. There is always some part of the human mind that is involved in a transmutation (otherwise, Mustang's gloves would only be able to mess with the air in a very select pattern), but in general more complicated transmutations will require a more robust "hard formula". The "soft formula" comes from the mind of the alchemist. This is the part that actually shapes what should happen, while the "hard formula" is used to gather the required energy/ handle the heavy lifting.
On the subject of people who can transmute without circles, there are two different schools of thought.
The first follows the anime/manga rigidly, and interprets Izumi's words literally. They see it as the 'reward' for losing part of yourself when you looked beyond the gate. Looking beyond the gate exacts a toll from you, which is then used to "pay" for the privilege of seeing the base code of the universe. Knowing this base code, alchemists are able to use the "soft" formula to a greater extent, and do not require the assistance of a "hard" formula in order to transmute.
The second possible explaination is more of an unintended side effect. Remember how Barry the Chopper lost his body? Well, his soul (in the armor) was able to sense his body's presence (and vice-versa). If one assumes the figure at the gate (which takes peoples' body parts) is god (or at least, the source of alchemy), then him "wearing" an alchemists parts (pardon the term) is essentially giving that alchemist a very subtle, yet powerful connection on a mental level. The alchemists are able to use their missing parts to bridge the gap separating them from this god-like being, and thus transmute seemingly at will.
Alphonse, not having a physical body at all with which to form a bridge, is unable to skip the "hard formula" at first, but upon re-visiting the experience in his own mind, he forces himself to build this "bridge" and can transmute naturally from then on.