Does quantum mechanics play a role in the brain? I'm interested in whether the scale of processes that occur in the brain is small enough to be affected by quantum mechanics. For instance, we ignore quantum mechanics when we analyze a game of tennis because a tennis ball is much too large to be affected by quantum mechanics. However, signals in the brain are mostly (all?) electrical, carried by electrons, and electrons are definitely 'small' enough to be affected by quantum mechanics. Does that mean the only way we will be able to further understand how the mind works is through an application of quantum mechanics?
 A: Quantum mechanics has almost no bearing on the operation of the brain, except insofar as it explains the existence of matter. You say that signals are carried by electrons, but this is very imprecise. Rather, they are carried by various kinds of chemical signals, including ions. Those signals are released into a warm environment that they interact with over a very short timescale.
Quantum mechanical processes like interference and entanglement only continue to show effects that differ from classical physics when the relevant information does not leak into the environment. This issue has been explained the context of the brain by Max Tegmark in The importance of quantum decoherence in brain processes. In the brain, the leaking of information should take place over a time of the order $10^{-13}-10^{-20}$s. The timescale over which neurons fire etc. is $0.001-0.1$s. So your thoughts are not quantum computations or anything like that. The brain is a  classical computer.
A: The brain is a de-facto a classical computer as explained in Alanf's answer. However, this then leaves open the possibility that what makes a classical system like our brain or some future AI conscious, could well be related to how quantum mechanics reduces to classical mechanics. One proposal along these lines (that I personally don't find compelling) has been put forward by Roger Penrose as mentioned in Ed Yablecki's answer.
A far simpler idea is to consider that quantum mechanics in the classical regime is still not the same as classical mechanics. What happens is that due to entanglement with a huge number of environmental degrees of freedom, many typical quantum effects are effectively lost and you can then pretend that they don't exist. As far as predicting the outcome of experiments is concerned, you can use classical mechanics with impunity. But the physical system is not what you get when you take its classical description to be literally correct.
You can see clearly how the difference between the exact quantum mechanical description of an AI + environment and the classical description of the AI answers a lot of the philosophical objections against the strong AI hypothesis. In the exact description there is plenty of room to invoke correlations between inputs and outputs as existing at any particular moment, because what the AI experiences is only a coarse grained measurement that is consistent with a large number of microstates. These then exist as parallel worlds within its de-facto measurement error. The real existence of such an ensemble of correlated states defines which computation is actually being performed at any given instant. The difficulty of doing that within a purely classical picture is at the heart of the criticism of strong AI.
Consider Marvin Minsky's famous thought experiment of simulating your brain by a huge analogue device consisting of huge wheels and gears. Then strong AI says that this simulation will succeed, but the critics say this is just ridiculous, how on Earth can a collection of wheels and gears feel anything at all? They key observation to be made i.m.o. is as follows. From your point of view, the exact state of the wheels and gears cannot be pinned down precisely. While you can look down on several of your wheels, any attempt by you to find out the state of all your wheels will fail due to your memory having a finite capacity; most of that capacity is used to run the programs that define you. Whatever you feel, whatever consciousness really is, it is ultimately a computation and a system of wheels and gears can define that unambiguously, provided you invoke the MWI.
A: In fact electric charge is carried by positive ions (sodium and potassium), not electrons, along the neuron's axon. Are they small enough? I don't really know. By the way, there is a hypothesis that a certain species of birds use a pair of entangled electrons to orientate themselves. Quantum mechanics also plays a role in enzymes (quantum tunneling for example) but due to the large amount of enzymes I don't think quantum effects make a difference. There is a good video that shows some hypothesis of how some organisms make use of quantum mechanics (it's not specific to the brain, though).
As for the brain itself the answer is no one knows, maybe it's used to store information or who knows what. As you said it'd be a good thing to take quantum mechanics into account (sometimes at least) to try to understand more of the brain.
A: Yes - but only in the sense that all macroscopic processes depend on underlying quantum mechanics at the microscopic scale.
No - quantum mechanics is not the best model for describing what happens in the brain.
In one sense, the behaviour of a neuron is similar to a quantum process, such as (for example) the decay of an electrically excited or radioactive atom to its ground state. A neuron either fires or it doesn't. But there are many machines that either fire or not, so this is not sufficient for us to infer that this a quantum process.
https://en.wikipedia.org/wiki/All-or-none_law
There are some important differences as follows (the most important of which is the scale of the process.) 
The atom emits a photon (a single quantum of electromagnetic radiation) randomly, and independently of events in its environment (for spontaneous emission, at least). We can know experimentally what the probability of a particular type of atom emitting a photon in a particular time period is.  
The neuron emits a an impulse (a large number of ions) in a fairly predictable manner, depending on the impulses and stimuli it has received. A good (if rather basic) model of this would be a water tank that automatically empties itself when it is completely full. Such tanks are used to flush urinals in mens' public bathrooms. It's a big step from this to building a computer as sophisticated as the brain, but it should be clear that such a tank does not depend on quantum mechanics. Note that an electronic analogue of such a tank is possible.
A: There are definitely some serious scientific efforts going on today trying to explain and incorporate QM in brain processes.  The following TED talk is just about that:
http://www.ted.com/talks/jim_al_khalili_how_quantum_biology_might_explain_life_s_biggest_questions?c=922691
Other scientists hypothesize that the tiny dendrites or microtubules on neurons inside brain are the interface where the feeble quantum effects take place and create the known classical effects.
The Physician Mark Germaine writes in his essay:  “A wealth of data supports the notion that the dendritic arborizations are the primary structures that support perception (Pribram, 1991). The neural wave form characterizes the dynamics of the dendritic network, and this wave form can be described by an equation that is fundamentally the same as the equation describing the quantum wave form (Pribram, 1991),………”  
But he is discussing other model and not directly the QM model for brain.  Find the essay here:  
http://dynapsyc.org/2015/HOLOMIND.pdf 
The point is that the commonly accepted mechanistic model for brain does not look as being enough to account for the entire richness of experience – not even the seemingly most basic ones like taste, smell, vision.  The mechanistic model cannot account for the enormous versatility of any consciousness – not even as simple as of an ant.   
A: Roger Penrose and Stuart Hammeroff are working on this exact hypothesis.  They believe the spindle fiber is the structure that collapses the Quantum Wave Function. As of now, they have been unsuccessful in showing the "spindle fiber" in supporting QM abilities. But there is plenty of info on the subject, starting with Roger Penrose book "The Emperor's New Mind".
A: I'm late to the party but didn't see the following argument, which is at the intersection of ethics, physics and logic. It has to do with the free will which we perceive we have.
The logical definition of free will is that it is not determined; that's the meaning of free. We do something not based on rules, i.e. we do it not predictably, but spontaneously. We could as well have decided differently, but we didn't.
In general, we do not see predictable behavior as signs of free will: Such behavior is guided by instinct, morale, conventions or external influences like advertisements or peer pressure. 
Signs of free will, by contrast, are visible at cross roads: Somebody decides to be a hero or not, eat at one restaurant or another. We, and possibly the person herself, didn't know beforehand.
Generally spoken, all behavior falls in one of these two categories: It is rule based, i.e. predictable; or it is a decision out of our free will, which nobody could reliably predict.
The interesting thing is that from a computational standpoint, "unpredictable" is simply equivalent to "random". That's the definition of "random": It does not depend on previous events, i.e. there are no rules to predict a random event from previous events.
Free will is logically the ability to make random decisions.
(As an aside, this is the reason that there will be no categorical obstacle to emulating human behavior with computers. All behavior is either rule-based or random. We can emulate rule-based behavior very well with computers. But it's also not that hard to introduce or emulate randomness.)
Let's not fool ourselves. Much of what we perceive as free will is not free at all; the overwhelming majority of our behavior is governed by our culture, ethics, taste, principles and so on, often unconsciously. That we seem less predictable than we actually are is owed to a lack of information on the side of the observer (and big data is telling us that we do become fairly predictable -- and gullible -- given enough information about us).
But we do have a strong feeling -- and the occasional examples -- of personal freedom. If we were mechanical, deterministic clockworks, we wouldn't be free. But we are not clockworks; the working of our brain is not entirely deterministic. 
The physical underpinning of this indeterminacy must be random events in our brain. Electrons, atoms and molecules are not billiard balls; no Laplace's demon could predict a brain's future, principally not. The microscopic world which is the underpinning of the world we perceive is simply not deterministic. In a non-linear system as the brain small quantum events which could have gone either way will occasionally make a difference. A neuron fires, or not; a group of neurons' excitation just crosses the threshold in a contest with another group to become dominant, or not.
In this sense, as a source of innate indeterminacy, and thus freedom, I believe quantum effects play a huge role in our brain. And in the universe.
A: Brain as a biocomputer is far to complex to expect an fully descriptive answer in a form of a forum post.
Don't worry, the situation is even more riddled! :) The opinions on the topic of quantum behaviour affecting the perception of reality (and accordingly creation of "reality", but I am touching very metaphysical viewpoint here, which is usually avoided here, and for good reasons) vastly differ.
All in all, in my opinion the currently correct answer is "Due to complexity of the brain, we do not know, where the quantum phenomena apply and where not in terms of brain and perception." At this point, one has to join the research to start getting at least partial answers.
"(...) I don't feel frightened not knowing things, by being lost in a mysterious universe without any purpose, which is the way it really is as far as I can tell." R. P. Feynmann
