What is "Qi power"? Has it been scientifically investigated? How can we achieve this?
It is the idea that you can physically use spirit energy. If you've ever watched Bleach (you should if you haven't. It's awesome), you'll hear them referring to it as reiatsu.
It's totally a myth of course. What old school practitioners of martial arts understood as spirit energy is actually just good old physics and/or body mechanics. When "building up qi", all you're really doing is relaxing your muscles which means there is more "snap" in your technique. Or when you "use qi to root yourself into the ground", all you're really doing is acting like dead weight. If you've ever tried to pick up a friend when his muscles are tense and again when his muscles are limp (dead weight), you'll understand the difference.
You can improve your "qi" by increasing your fast-twitch muscles.
Study on acupuncture: http://consensus.nih.gov/1997/1997Acupuncture107html.htm There's also this hilarious fight between what seems to be an MMA fighter and a chi master: http://www.youtube.com/watch?v=djUKqxGWj_Y
What do we mean when we say Qi?
It's not possible to scientifically study Qi until people agree on what it is. If it's good body mechanics, then that's perfectly reasonable. Nobody would argue that some people use their body well and can punch hard because they know how to relax and efficiently use their muscles.
But if Qi is supposed to be something supernatural, then it very quickly becomes absurd. People claim they can perform no-touch knockouts with Qi, but they always fail when asked to do it on someone who is not their student. That tells us that their students are playing along--usually due to unconscious social pressure. Note that in that video, Dillman says that Qi is a radio wave. If it's a radio wave, then we can test that, study it, see if it's present, and examine where it comes from. In fact Qi is not a radio wave--Dillman is just making up BS.
Qi debunked--over and over and over
Proponents of Qi will always find a way to keep believing, despite every instance of Qi being shown to be bunk. Skeptical people keep testing claims of Qi and keep finding them to be plainly false. Over and over, whenever we scientifically examine Qi, it turns out to be a parlor trick. Scientists aren't interested in claims so vague that we don't even know what Qi is or what it does. To scientifically study it, first decide what you're trying to prove:
Qi is similar to the medieval idea of humours, or astrology. They are both clumsy attempts to describe very real phenomena (people getting sick, people getting knocked out with a punch to the neck) with naive and ultimately incorrect frameworks. It turns out that the planets don't even move the way that astrology describes, and that phlegmatic humours were just a shot in the dark that turned out to be wrong, and that Qi was just a way to vaguely describe medical and martial phenomena that people didn't understand.
The word "Energy"
It's perfectly reasonable to talk about energy in the context of martial arts. Wrestlers and mixed martial artists use it to simply mean "resistance" or "oomph". It's silly to think that the word is anything more than a metaphor or out-of-context hand-waving. It doesn't mean anything like what scientists mean when they talk about electrical energy. It's silly to claim that the word relates to any specific, tangible, yet mystical force.
I think the main thing Qi does is subconsciously boost your own willpower. This may be simplified, but it works like a placebo. If someone believes they have an increased energy about them that will allow them to defeat their opponent, their thought goes less into fear and doubt, more into trusting their body and their knowledge/technique.
When training at home, i find that my spiritual excersises from tai chi have absolutely no effect on my performance. At first i relied on these techniques before training, but now i know the precise way to use each muscle and joint, i do not waste my time.
Being that most people seem to be incredibly biased on the subject, i doubt there is any credible scientific work into the subject. As for 'How can we achive this?', have no doubt in your technique and train for a lifetime. Keep in mind the martial arts grandmasters who claim to use Qi have been training their entire lives and have the skill and willpower to outmatch someone twice their physical strength.
Your Qi or Ki or Chi or Energy or Willpower is really how much energy [as in actual electrical charge] you are willing to put into your body's muscles before deciding that you have lost and will not win the arm wrestle, or will not be able to hold your ground when being pushed, i could go on.
One more thing i have to mention, on my first kung fu lesson, i watched a video of myself, my brother and two friends [with our eyes closed] being pushed forward or backwards by the sifu without him touching us, we were told this was Qi. His hand was very close but not close enough to make contact with the skin or hairs so i think this is a combination or two things, - the fact you are expecting to be pushed in one direction -your mind picking up signals such as heat, sounds at a negligible amount you would not normally pay them any attention and/or [i would like to believe, because it's cool] minute interference in your electric circuits due to the radiation given off by another persons electric circuit in close proximity. Either way it is nothing supernatural.
I used to believe in Qi when i started martial arts until i realized it was, as i say, a placebo.
An interesting article on how Qi came west
A few chinese (born and trained) tai chi guys I've worked with have said "When the old people talked about Qi they meant leg strength".
Qi is a term that gets used to mean a lot of things, and the most common usage as an independent energy force, is regularly shown to have no evidence supporting it. That said, here's some ideas of the different things people mean when they talk about qi:
Muscle & Joint Activation
"Try to move my arm! You can't! My qi is what keeps it up!" A lot of these are good uses of body mechanics - often positions that allow you to use both agonist and antagonist muscles together for structural efficiency in holding static positions, or intelligent use of positioning and muscle chains for dynamic movements.
Activating some of these muscles often works best with a bit of visualization or movement that seems completely mental - "Hold your arm out, point at something in distance, try to reach it with your fingers, as if you were extending your energy beyond your fingers - seen how strong your arm is now? See how hard it is to pull it down? This is Qi."
Although thinking of energy flowing to different parts of your body, or beyond your body can help you do stuff like this, you can replicate the same results with good body coaching in other terms, so it's not any kind of energetic force.
The flipside, especially used in martial arts, is that you can "feel their qi by touch" which mostly is reading the intent of the person's movements because you can feel subtle changes in their balance and muscle tension - basketball players do this all the time when they stick someone.
Assessment of several body functions
If you end up studying acupuncture or tradition chinese medicine, they may use the word qi, but you find out there's over a dozen qi types referred to in the body: "circulatory qi", "Heart qi", "digestion qi" etc.
In this case, when you assess someone and try to figure out how to help them, the quality of the individual qi is looking at the combined results of several body functions working together - your digestion for example involves your stomach, gallbladder, liver, small and large intestine, etc.
In this sense, it's basically clumping a lot of organs together into a larger system, and it's not something to prove or disprove anymore than it's a convenient way to assess overall health, much in the same way doctors say "Circulatory system" without necessarily talking about each part.
Many translate qi as "wind" or "breath" so it's not a hard guess as to why this is the way it gets used sometimes. No breathing = dead, breathing = alive, so it's not hard to see how a lot of folks would see this as the sign or source of life energy. An old Chinese folk idea is that everyone has a limited number of breaths they get before they die in life, so slowing the breathing extends your lifespan.
This is the usual method talked about, which involves the usual point of qi pathways, acupuncture, acupressure, and even action at a distance. So far, no scientific study has proven or matched 1:1 to traditional Chinese medicine theory, martial arts theories, or meditation/yoga type work in this manner. Several studies have reported minor but notable benefits for several things (pain relief, increased mobility of limbs, lowered blood pressure), but nothing has matched completely.
How can we scientifically study this?
Well, lots of studies are being made all the time. Including in China, Taiwan, Singapore and many places where TCM is already a major part of the society.
If you want to study outside the medicine side, I would be looking first to understand explicitly what you're trying to get from it, since, with so many definitions, and people often using qi for any/all of them, you would want to structure your experiments to test for just one of them.
A secondary and useful context is to also understand that anyone advocating a skill or ability with qi is probably extremely varied in terms of skillfulness and training. Outside of China's TCM system, which is a fairly modern construction, everyone got individualized training of different theories, quality and ability even within the same "school" or lineage.
And a lot of these things also underwent a lot of destabilizing factors that made it difficult to assure transmission - look up the history of civil war and famine in 20th century China where 40 million people dying in a short period is kind of a regular occurrence. Stack on top the Cultural Revolution driving a lot of people out who have these skills or schools and you can see even people who claim the same lineage may have very different theoretical understandings and practices of qi.
So I would probably add a third layer, which is even after you narrow down what you're testing for, I'd probably also sort out the people you'd be testing based on what they claim their theory base is of qi - since it may drastically affect what they're really doing and what outcomes are the result.
Many of the other answers discuss the rational Western view of qi (or chi or ki)--that it is a mythical "wind," "power," "spirit," or "energy" claimed by traditional Chinese medicine, Daoist cosmology, and traditional martial arts heritage to be the primary invigorating force in human beings. It's a popular idea, with analogs across many cultures. It's the "ki" in Korean or Japanese arts. It's echoed in yoga and elsewhere.
The rational, Western, scientific view is that it doesn't exist. It's a concept inherited from an ancient world that did not understand how physics, biology, and medicine really work, and so invented a poetic, shamanistic, religious story as though they did.
As a rational Western scientist, I don't disagree with this skepticism.
Whether you believe qi is a "real," tangible thing/force may not be the most important issue.
My experience is that training as though qi is a thing is genuinely helpful even if you don't believe it's real. How can that be? Because considering, talking about, and training qi focuses attention on useful aspects of what we feel, how we physically act, and what we intend to do that we would not otherwise be focusing on.
In training marital arts, acrobatics, dance, and other motion-involved activities, many of the elements of kinesthetics, balance, and proprioception are subtle learned capabilities--things we must learn to do, feel, or feel comfortable with that are not necessarily "built in" or entirely natural. Considering qi can focus our attention on gathering and emitting energy--a focus that is functionally useful, even if it's only a metaphor.
There is actually a lot more precedent for this type of model in Western thought than you might think. Legal fictions, for example, are places courts and legislatures assume a set of facts or circumstances that are not strictly true--because it's useful to do so. Philosophy--whether ancient or modern--is replete with such assumptions. And the clincher: So is modern science and mathematics. See e.g. electrical engineering, quantum mechanics and string theory, which eagerly use models of virtual and imaginary constructs when "the math works out better that way."
So be a skeptic about qi if you like, or outright disbeliever. I am! But that doesn't mean you shouldn't consider or use the metaphor as a martial arts (and motion arts) training tool. I do, with good success.
Let us try to combine the descriptions an experience of a 3000 year old culture with the scientific accievments of our time:
To my understanding Qi means the energy or a fighter has available. That is to say, if I am tired or exhausted my Qi is low, but when I am well rested I have a lot of Qi.
That would fit the answers others have provided like "Qi is the power of your kick".
It also would also provide a scientific prove that greter Qi can be attained trough a healthy life style and MA practice, the latter allowing you to create more powerful techniques through more precise execution.
Also certain breathing techniques are known to calm your body, thereby removing stress, thereby allowing a regeneration and increased energy.
Finally, from an ethymological viewpoint, I the last syllable of "energy" sounds a lot like "Qi" (Chi).
Qi is your life force energy. When you die, your life force leaves the body and the skin and flesh become whitish in hue.
You Qi is strengthened in a number of ways (I'm only a piker here.). The breathing methods in traditional karate (as well as other martial arts, yoga, etc.) are a primary way of building up your Qi. Diet, rest, and exercise are also elements of building up strong Qi. Again, I have only a layman's understanding.
The martial arts applications of Qi progress from increased health, to increased physical strength, to levels where the practitioner can mentally direct Qi to areas of the body for protection or to generate force by itself (advanced).
Martial arts exercises, starting with basic techniques, builds Qi through the proper breathing and alignment of the body, which then facilitates the circulation of Qi throughout the body.... leading to the enhancements described above....
Qi, as pure energy, is problematic to sense or measure. As one poster comments, empirical evidence of Qi can be seen in the pink color in one's palm's, the breaking of bricks and boards (especially good vid of the chinese doing this.), and the resistance of the body to what would be certain injury (chinese vids again).