The following text, which is under construction, is an introduction to quantum mechanics for the layperson. See Quantum mechanics/Advanced for a more technical exposition.
Quantum mechanics (from the Latin quantus, "how much") is a theory in physics that explains and predicts the behavior of matter and energy at very small scales — behavior which is often unusual and sometimes extremely counter-intuitive, since it is deeply in conflict with the ideas most people have of how the physical world works. It is perhaps the most important building block in the revolution of physics (1900-1925 period) which erased the limitations of classical physics and created the physics of today.
Quantum mechanics, and the understanding of quantum entities (i.e. things which operate under the laws of quantum mechanics) that it provides, has been an absolutely indispensable tool in the creation of much of today's modern technology. In particular, the entire semiconductor electronics field uses quantum mechanical principles — and without semiconductor electronics, the now-ubiquitous miniaturized and cheaply mass-produced electronic devices of today (such as computers, cell-phones and cameras) would be utterly impossible. Also, lasers and medical diagnostic tools such as MRI (magnetic resonance imaging) could not exist without a knowledge of quantum mechanics. Modern chemistry (and through it, biochemistry) is increasingly relying on the principles of quantum mechanics to further its understanding of molecular interaction.
Quantum mechanics is extremely important, and not only for the technology it has given us. What the scientists who uncovered quantum mechanics found was that many of the supposed fundamental principles that underlie how reality operates (e.g. causality, locality) are not fundamental at all. Simply put, the 'rules' appear to be absolute at the large scale in which we experience physical reality, but they do not exist when reality is examined at very small scales.
An explanation is that the 'rules' we perceive as governing the behavior of reality are often only due to large-scale statistical artifacts. To give an analogy of this particular aspect, if one viewed the result of flipping a coin a million times, one might gain the (false) impression that any time one flips a coin a given number of times, exactly half the time one will get tails, and half heads. This is of course not true if the coin is flipped only a few times: flip a coin four times, and on average, one eighth of the time one will get the same face showing all four times. 
It is these extremely counter-intuitive truths about how the physical world actually works that contribute to the difficulties most people have when they first encounter quantum mechanics. Indeed, it has been said the only physicists who aren't bothered by quantum mechanics are the ones who haven't thought about it. In making these discoveries, the discoverers of quantum mechanics have deeply affected our understanding of the very nature of reality.
Principal findings and predictions
Among the principle findings and predictions of quantum mechanics are:
- Light, and all electromagnetic radiation, is not emitted in a continuous stream of energy, but in very small units of predetermined energy, called quanta - from which the theory derives its name.
- There is a fixed relationship between the wavelength of a quantum of light, and the amount of energy it contains.
- Not only light, but also energy and a number of other fundamental substances (such as electric charge) are quantized, i.e. not infinitely divisible. Indeed, space and time themselves are generally thought to be quantized, too, although the details are still obscure. 
- Classical physics holds that light is comprised of waves, i.e., travelling perturbances in the electro-magnetic field, but light also (most paradoxically) appears to have characteristics of particles, i.e. entities which are of fixed size and form. It is for this reason that the quanta of electromagnetic waves are also called photons — i.e. light particles — using the -on ending traditionally reserved for particles.
- Not only do things usually thought of as waves have particle-like aspects, but things usually thought of as particles (e.g. electrons) also have wave-like aspects; this wave-particle duality is now seen as an inherent aspect of all quantum entities. 
- This raises the question: what is a particle anyway? The naive model, that it is something like a small ball, is clearly — once again — the result of the incorrect assumption that the world at the quantum level looks the same as it does at the level of reality we experience, only much smaller.
- Many processes at the quantum level are only seemingly deterministic; i.e. while their behavior, when measured in large numbers, follows some law (as in our coin-flipping example), individual events are not predictable. For example, given a large amount of a radioactive element, it is possible to accurately predict how many of those atoms will decay in a particular amount of time. It is, however, impossible to predict if, and when, any particular atom will decay.
- Quantum mechanics also appears to indicate that for many attributes of a quantum entity (e.g. its spin), that attribute does not have a fixed, definite value until it is measured. In other words, that attribute (or, to be precise, its value) in some sense does not exist until it is measured. This particular point has been a source of much debate since the 1920s, which continues to this day.
- A particle cannot have a well-defined position and simultaneously a well-defined speed; this is one aspect of the famous Heisenberg Uncertainty Principle. This principle states that certain pairs of physical properties (like position/speed or time/energy) cannot have simultaneously well-defined values. In addition, performing a measurement on a quantum system inescapably affects the system. The measurement of one characteristic of a quantum entity inherently affects the values of other characteristics of that entity. This is not due to a simple lack of subtlety in the design of experiments, but is a fundamental attribute of all quantum entities.
Although much of quantum mechanics is now much better understood that it was when it was first being uncovered, there are a number of issues which are still unclear, and the subject of much debate among scientists.
The "Schrödinger's Cat" question
This is probably the largest open question in all of quantum mechanics. It derives its name from a thought experiment posed in 1935 by Erwin Schrödinger, one of the key theorists in the development of quantum mechanics.
Fundamentally, the problem is that quantum mechanics itself does not draw any boundary line between the 'quantum world', the level of scale where the strange laws of quantum mechanics apply, and the 'macro world', the level of scale of the world we live in. There is no place where one can say 'this is where the laws of the quantum world fade away, and the laws of the 'normal' world take over'. As far as quantum mechanics is concerned, any system, of any size, can be described by the rules of quantum mechanics - and this can lead to situations which seem to make no sense, a point made most forcefully by the one posed by Schrödinger.
Most formulations of quantum mechanics operate as if a quantum system does not have a definitive state unless it is somehow measured (in itself, still something of a point of contention). In this view, if a single radioactive atom is isolated, the rules of quantum mechanics indicate that until the atom is observed, it does not have a definite 'decayed' or 'not decayed' state; rather, it is in some indeterminate state in which it is both at the same time. Schrödinger put this notion together with the observation that quantum mechanics can be applied to a system any size, to produce his problematic cat.
He imagined a closed box containing a cat, a vial of poison, a single radioactive atom, and a mechanism to detect the decay of the atom, and release the poison. The entire assemblage is a system which can itself be described by the rules of quantum mechanics - i.e. until we observe it, by opening it to see if the cat is dead or alive, it has no definitive state. The atom is neither decayed nor undecayed, the poison is neither released not unreleased, and the cat is... neither dead nor alive.
This is 'clearly' nonsensical - cats are either dead or alive, not somehow mysteriously both at the same time. However, there appears to be nothing wrong with the application of quantum theory which predicts that the cat is both dead and alive at the same time. Since quantum mechanics is one of the most successful scientific theories of all time, in that its (often bizarre) predictions have always been accurate when they were tested, this is very troubling. In addition to which, of course, quantum mechanics is full of all sorts of other 'impossible' things, so nobody is ready to write off the cat problem as 'impossible'.
To make things even more problematic, in recent years experiments have been done which show macroscopic objects (superconducting rings) which behave as if they were quantum objects. In other words, when the ring changes state (e.g. a change in an electric current flowing around the ring), the entire ring changes state at the exact same instant. 
So the possibility of bizarre quantum behaviour at large scales - including cats which are simultaneously dead and alive - is not utterly implausible. However, ever since the question was first posed, most physicists have not believed that the cat is actually in that intermediate state; most agree that 'somehow' the atom is 'observed' and does either decay or not decay. What nobody can do is explain how - in terms of the physics - this actually happens.
In the many decades since Schrödinger originally posed his problem, it has been vigorously debated by physicists, but an answer is no closer, or clearer, now than it was when he first posed it. Many (competing) theories and explanations have been put forward, but none has been proven, or even gained broad support.
Locality versus Reality
Strictly speaking, Bell's Theorem, and the experiments spawned by it, only mean we have to give up one aspect of the 'classical' model of reality. A 'hidden-variables' theory, where the universe has a real state rather than just a set of probabilities, can give the same empirical predictions as standard quantum theory, but only if it is 'non-local'. That is, influences can propagate faster than light.
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The discovery of quantum mechanics
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- Overview of classical physics (i.e. continuous energy, space, time)
- First clues
- Black body curve
- Radioactivity, although that's mostly nuclear physics
- First steps
- Planck equation/constant
- Einstein 1905
- Bohr quantized atom model
- Full glory
- de Broglie
- Born's probability
- EPR 'paradox'
- Bell's Theorem
Some unusual effects of quantum mechanics
Quantum mechanics produces some very unusual, and hard-to-believe, effects. This section lists some of them.
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- 3-filter polarization example (people can try this one themselves): 2 filters at 90° to each other, no light gets through; add a third filter between them, at 45°, now some light passes
- All the weird double-slit stuff
- Photons emitted one at a time still create interference patterns
- When you look to see which slit they go through, the interference patterns go away
- Macroscopic conducting rings acting as quantum objects
- Bell's Theorem, and experiments based on it, have shown that the nature of space, and causality, is very different than we (and Einstein) have understood it to be. Two particles created in a single quantum event appear to share some mysterious instantaneous connection, no matter how far apart they may later travel. One particle 'knows' instantly when some important change happens to the other particle. The implications and technological possibilities of this relatively recent discovery are still being uncovered today.
- One of the consequences from this theorem is a quantum teleportation — you can transfer the quantum state from one place to another without any physical contact between them besides transferring some information (so you cannot exceed the speed of light in this process). An interesting fact is that the quantum state must be destroyed in the first place — arbitrary quantum states cannot be cloned.
- 'Locality' means that an action cannot have an instantaneous effect at another location a very great distance away. No less a physicist than Einstein appears to have been tripped up by this particular incorrect belief; "No reasonable definition of reality could be expected to permit this" is a well-known quotations from a now-infamous paper he co-authored with Boris Podolsky and Nathan Rosen in 1935.
- That is, 1/16th of the time one gets all heads, and 1/16th all tails, so 1/8 of the times the same face shows.
- Is time quantized?
- Macroscopic objects are also believed to show this wave-particle duality; a glass has wave characteristics, but its wavelength — which can be easily computed by the laws of quantum physics — is too small to be of any possible significance.
- Strictly speaking, Bell's Theorem only makes this certain for 'local' theories, i.e. theories in which information about actions in one place cannot be instantly transmitted arbitrary distances. Bell's Theorem, and its exact meaning, are still a source of considerable debate to this day.
- R. J. Prance, A. P. Long, T. D. Clark, A. Widom*, J. E. Mutton, J. Sacco*, M. W. Potts, G. Megaloudis* & F. Goodall*, School of Mathematical and Physical Sciences, University of Sussex, Macroscopic quantum electrodynamic effects in a superconducting ring containing a Josephson weak link, Nature 289, 543 - 549 (12 February 1981)
- Dennis Overbye, Quantum Trickery: Testing Einstein's Strangest Theory, New York Times, December 27, 2005 - A report on a recent experimental result serves as a hook for an engaging and accessible discussion of some of quantum mechanics' stranger aspects and implications.
- John Gribbin, In Search of Schrödinger's Cat: Quantum Physics and Reality - Although now somewhat dated, it remains one of the best introductions to the strange world of quantum mechanics for the non-scientist.
- P. C. W. Davies, J. R. Brown, The Ghost in the Atom: A Discussion of the Mysteries of Quantum Physics, Cambridge University Press, 1986 - A somewhat advanced book on the subject of the EPR 'paradox', Bell's Theorem, and the experiments which if fostered: it contains interviews with many of the scientists involved, most of whom contradict each other!
- N. David Mermin, "Is the Moon there when nobody looks? Reality and the quantum theory", in Physics Today, April 1985, pp. 38-47. - One of the most comprehensible explanations of the original EPR debate, Bell's Theorem, and the Aspect experiments, and what it all means.
- N. David Mermin, "Spooky actions at a distance", in Boojums All The Way Through, Cambridge University Press, 1990, pp. 110-176. - A somewhat longer article covering much the same ground.