The underlying assertion of most goofy new-age claims about quantum mechanics and the brain is that consciousness is a quantum process. Of course, in a trivial sense it is quantum insofar that every process in the physical world seems to obey quantum mechanics. The exact claim is that something "essentially quantum" is behind the phenomenon of consciousness, that the computations of the brain actually exploit uninuitive quantum behaviours that cannot be explained by a classical physics picture -- the claim is that our brains are quantum computers.
You build a quantum computer by exploiting the fact that a simple, perfectly isolated physical entity does not act like a tiny billiard, but rather as a complex-valued wave that isn't in any particular place at a given time, it's spread out. We say that small systems can be in "superpositions" of multiple states. Now when the system interacts with the environment, by hitting a photon from our lasers, say, it will "collapse"into one state, we will see the photon bouncing off as though the particle had been at one particular place.
Quantum computers exploit superpositions by encoding data into the states of small particles and allowing them to interact and evolve with each other isolated from the environment in such a way as to perform a simple calculation without collapsing until the very end when you measure (read out) the answer. The advantage comes from the fact that every possible history of the simple computation is performed in such a way that certain parallel algorithms can try all the combinatorial possibilities at once before being summoned to give an answer when they're collapsed. A quantum computer with N bits is like classical computer with 2N bits.
The key requirement is that the quantum computer not interact with the environment (stray light, cosmic rays, etc.) during the duration of the calculation. The more complex the computer, the larger the number of particles needed to encode the data, the more exquisitely sensitive the computation becomes to outside noise:
It is so difficult to get more than a few particles isolated long enough to perform calculations that after about 10 years of effort, the biggest quantum computer has about 8 qubits. The quantum brain hypothesissays that there is some remarkable way that the proteins of our neurons could form an isolated network of qubits such that the brain could perform quantum calculations, and that the mysterious nature of consciousness could be chalked up to the weirdness of its quantum underpinnings.
There are two major reasons for why this quantum brain hypothesis is extremely doubtful:
Any time a stray particle hits a quantum computer it collapses back into a world-entangled state, ruining the computation before it's finished. The brain is a hot, disordered, massively chaotic place with countless particles bouncing into everything a billion times a second. The longest biological quantum superposition known happens in chlorophyll, and that lasts about a trillionth of a second.
A general, conservative calculation about the survival times of quantum states in the brain was done by Max Tegmark , suggesting a trillionth of a second as the limit. Now, biology certainly does exploit quantum effects at the timescales at which they happen in cells. i.e. quantum effects influence photosynthesis and the electron transport chain of respiration. Natural selection cannot select for life that uses extremely short-lived physical processes to perform long-lived tasks.
Mental phenomena seem to be explicable solely in terms of the electrical spikes neurons use to signal with each other. i.e. In animal experiments, we can see that the external information of sight and sound is encoded in the frequency and pattern of these spikes in populations of neurons. Controlling these electrical signals artificially seems to influence animal behaviour in a predictable fashion.
We are still very much in ignorance of the brain's operation, but not so much at the level of its biophysics. More so in the level of detail about how these signals work across the hundred billion or so neurons of the brain, and how individual neurons alter their connections and sensitivities over time to other neurons. The fastest of these processes happen at the millisecond timescale, meaning that any quantum process is much too fleeting to influence the phenomenon we know to be directly involved in neural computation, by a factor of at least 109!
The consensus scientific opinion is that the brain acts as a massively parallel, stochastic,and classical computer. Whatever the "secret" to consciousness is, it's not quantum superposition.