Quantum mechanics is notoriously weird and difficult to comprehend. Many people I speak to seem to know of it, but not much about it at all- which is my main motivation behind writing this article. I will (briefly) go through some of the insights that quantum mechanics has provided about the way we see the world around us, and hopefully clarify things- at least so that the next time the word ‘Quantum’ is used somewhere completely out of context, you’ll be able to call ‘Bullshit’ on whoever is trying to sound like they know what they’re talking about (happens a lot more than you’d think!).
The term ‘quantum’ was coined by German physicist Max Planck (1858–1947) to describe a discrete amount of energy- a ‘package’ of energy. Planck, who was one of the founding fathers of quantum theory- was researching the relationship between intensity (the amount of light) and frequency of light. At that time, there were only empirical laws to describe this relationship (Wien’s Law for high frequencies, and the Rayleigh-Jeans law for low frequencies) but no solid theoretical framework which could successfully make predictions in accordance with these .
As part of the solution, Planck postulated that energy can only occur in discrete packages, or ‘quanta’. From this postulate he was able to derive a law which stood in accordance with experimental data, and therefore with the empirical laws known at the time.
This short, but sweet equation gives one the relationship between the energy (E) associated with a certain frequency of light (f). h is known as Plancks constant, which is always the same, tiny quantity. This means that the value of h*f is always the smallest package of energy attainable for a given frequency.
However, it must be noted that Planck merely postulated this thinking; it was a mathematical trick which happened to yield promising results- It did not yet occur to him that he had fundamentally changed the way physicists would view the world. Five years later Albert Einstein used Plancks postulate in order to provide a theoretical understanding of the Photoelectric effect- A current generated in a conductor, through illumination with visible light/ Ultraviolet. This work gained Einstein the Nobel Prize in physics in 1921 (contrary to popular belief Einstein’s work on Relativity, which he is most famous for, did not win him a Nobel prize) .
From the definition of the quantum sprang a whole new branch of physics which we now know as Quantum mechanics. Before getting carried away with the fascinating history of quantum mechanics, I will focus on a few key ideas which have emerged from it- ones I often find myself speaking about when people ask ‘could you just explain quantum to me?’. I will certainly do my best. I want to note that this article will not explain how these phenomena work, but rather lay out different phenomena and present some evidence for these occurring. I will, however, provide several links to papers which attempt explain how to the most interested of readers.
Wave- Particle Duality
For a very long time scientists were locked in a heated debate about the nature of light. Scientists such as Christiaan Huygens believed light to be a wave, because of the many wave-like properties exhibited by light, such as diffraction, polarization and interference. However, Isaac Newton believed light to be composed of particles- a theory which was mainly popularised through Newton’s prominence. It wasn’t until the early 20th century, when Albert Einstein successfully explained the photoelectric effect using particles of light (photons) that the particle theory of light was seriously considered again. 
What quantum mechanics has come to show is that light, as well as all other fundamental particles behave both as a wave, and a particle depending on the circumstances. Lewis de Broglie postulated that particles of matter are also waves of some sort, which have an associated wavelength (which is known as de Broglie wavelength).
There is an overwhelming amount of evidence to support this[6,7], and in recent times wave- particle duality has been extended far beyond photons and electrons. Wave- particle duality has been demonstrated in objects such as large organic molecules - forcing us to reconsider the definitions for ‘particles’, and ‘waves’ and to ask ourselves the question whether anything ever exists as purely a particle, or a wave.
Uncertainty is the biggest factor setting quantum physics aside from classical physics. In fact, this is an extension of wave-particle duality since the phenomenon of wave-particle duality only arises due to a particles uncertainty in position; allowing it to exhibit wave like properties. Before going any further I would like to make a distinction between two types of uncertainty, which both play a big role: uncertainty arising from our inability to know something (information inaccessible to us due to a variety of reasons), and inherent uncertainty which exists within nature itself. Yes, as it turns out nature itself can be uncertain about its own properties- and this has nothing to do with our ignorance, or our method of probing. The mathematics in quantum mechanics is full of uncertainty relations; inequalities which restrict the amount of information obtainable about a system. Take for example the most famous of them all- Heisenberg’s uncertainty relation, which can be seen below.
Where ΔX is the uncertainty position, ΔP is the uncertainty in momentum and ħ is the (reduced) Planck constant which we have seen above (kind of). This means that the left hand side of the equation can never equal zero (in fact, never be smaller than ħ/2); there will always be uncertainty in the position, and the momentum of a given quantum particle. What this means in physical terms is that you can never exactly know the position and momentum (speed and direction) of a particle at a given time.
How can this be possible? Surely there is something that we’re misunderstanding? One attempt to explain uncertainty in a classical sense was to suggest that this was merely another form of the ‘observer effect’ . This effect is a consequence of the inevitability that the properties system cannot be measured without being altered; take for example measuring the temperature of a hot water bath. When introducing a thermometer, the temperature of the bath will be (very, very slightly) altered because energy has to be taken from, or given to the system in order to record a temperature.
The same reasoning can be used for measuring the position of an electron. If you want to know the momentum of the electron more precisely, you will have to use more energetic light (higher frequency) which, in turn, will change the momentum of the electron.
Even though this effect does take place, it is distinctly different from the quantum uncertainty being discussed.
Without making things a lot more complicated there is little I can say except for the fact that uncertainty is inherent in nature, and that we have more than enough evidence for this to be credible. In fact, some of our technology even utilises uncertainty, and would not function without it- an example being the scanning tunnelling microscope (STM)  and some forms of touch - screens.
The scanning tunnelling microscope makes use of the phenomenon quantum tunnelling. (Quantum) Tunnelling is a phenomenon in which particles can appear on the other side of an energy barrier, without having the necessary energy to make it over the barrier.
A classical analogy to this would be: picture a ball rolling from side- to side in a valley. It does not have the necessary energy to make it to the top of either side of the valley. Quantum tunnelling would allow this ball to spontaneously appear on the other side of the valley, without ever having the necessary energy to make it to the top of one of the hills.
However, tunnelling is simply yet another amazing consequence of uncertainty; because uncertainty also applies to the kinetic energy a particle possess.
A good and more rigorous explanation of tunnelling and its use in touch- screen technology is found here:
The phenomenon of superposition is one where particles tend to have multiple, seemingly mutually exclusive properties at the same time. For example position; besides there being uncertainty in a particles position, a particle can occupy numerous positions at the same time. This, again, is not a desperate attempt at explaining something that we can’t, but rather a phenomenon demonstrated time and time again .
Superposition arises straight from the heart of quantum mechanics- the Schrödinger equation. This equation is the quantum equivalent of Newton’s equations of motion; it governs the dynamics of a quantum system. Just like any other equation you input certain parameters, and then solve for possible solutions- these solutions being properties of the system. As it turns out, there are an infinite number of solutions to the Schrödinger equation- more precisely the sum of any solutions to this equation is yet another solution! This means nothing less than a particle exhibiting an infinite number of properties at the same time. What makes matters even more mind boggling is the fact that when scientists attempt to measure particles being in several states at once- they never do! Quantum objects start behaving classically when we measure them; one way we know that superposition exists is through secondary effects- outcomes which can only occur if two things happened at once.
Superposition has been demonstrated in countless experiments, such as the famous double-slit experiment. If the reader is not familiar with this experiment I would strongly advise to look it up- it is mind blowing!
Here is a link to a video explaining it quite well (but it’s slightly cheezy!):
Another more recent experiment placed not a tiny particle, but a nano-sized ‘tuning fork’ into a superposition of states; researchers got this mechanical object to vibrate at several frequencies at once!  Furthermore superposition is utilised by plants in their photosynthesis process making it up to 99% efficient!  (compared to 25%- 30% for pertol engines , and about 22% efficiency for solar panels ).
Surely the biggest application of the superposition phenomenon is the development of quantum computers; which famously use ‘qubits’ instead of ordinary ‘bits’. Whereas a classical bit of information is only ever a 1 or a 0, a qubit can also be in the superpositioned state of being a 1 and 0 at the same time. This is because qubits are made up from something exhibiting quantum properties; ultracold atoms/ions, photons, or currents in superconductors. This leads to vastly greater computing power because numerous solutions can be processes/calculated at the same time- once successful quantum computers will undoubtedly change the world.
Superposition applies to many different properties; whether its position, momentum, or even the chronological order of events! 
That’s right, it has been recently shown that not only can a particle be in several places at once, but that there are quantum systems in which things happen both forwards, and backwards in time! Astonishingly, this does not violate any causal inequality because these events happen in both directions of time- not just backwards.
It seems quantum mechanics is still getting weirder about 100 years after its initial formulation!
Entanglement is a phenomenon famously called “spooky action at a distance” by Einstein because it seemingly violated one of physics’ most sacred laws: the speed of light being the maximum achievable speed in the universe.
A set of entangled particles influence one another instantaneously, regardless of their separation. These particles could be on opposite ends of the universe, and this would still hold true. The conundrum is that seemingly this implies that information travels between one particle, and another at faster than the speed of light (something Einstein had shown to be impossible). As amazing as this is, the paradox is resolved through interpreting the situation slightly differently - something I definitely do not have time to explain in this article (for those particularly curious I can suggest the paper, entitled “Quantum mysteries disentangled” to which I have added a link at the bottom of this page)
Where is this ‘quantum’ and why don’t I see it?
The short answer is: because you’re looking.
For my university dissertation I simulated an experiment demonstrating that quantum mechanics naturally gives rise to classical physics- which we observe in our day to day life. When fully quantum particles (with all their weird properties) interact with their environment, they start losing their quantum properties. With environment, I mean anything.
Since our universe is filled with particles, constantly bumping into once another, interacting- one could say the universe is constantly measuring itself. As we have learned: measurements get rid of quantum properties, which is inherently the reason we see a classical, and not a quantum world.
This loss of quantum properties has been coined ‘Decoherence’ and is an area of active research - since it is one of the biggest obstacles facing quantum computers. In order for qubits to be 1’s and 0’s at the same time these qubits have to remain quantum, and not ‘decohere’. This means isolating them from their environment; which is proving to be extremely difficult.
It is very difficult to say anything more about decoherence, or the nature of quantum mechanics without going a lot further into detail, and writing a book about it which is why I would suggest to anyone curious or interested to start researching this stuff. I am aware that some of these explanations, to those more knowledgeable, may be rather simplistic- something I couldn’t really help, due to the sheer quantity of information I would have to provide to continue making sense.
As promised here is a link to the paper which attempts to explain how entanglement works:
And to those who are still unconvinced, I am happy to send my dissertation to upon request, in which I simulate this quantum to classical transition mentioned above, and explain the how of things in a lot more detail.
 - M. Planck (1914). The theory of heat radiation, second edition
 - Kragh, Helge (1 December 2000), Max Planck: the reluctant revolutionary, PhysicsWorld.com
 - Folsing, Albrecht (1997), Albert Einstein: A Biography
 - Feynman, R.; QED the Strange Theory of Light and matter, Penguin 1990 Edition
 - Darling, David (2007). “Wave–Particle Duality”. The Internet Encyclopedia of Science
 - Davisson–Germer experiment “The diffraction of electrons by a crystal of nickel
 - http://www.livescience.com/19268-quantum-double-slit-experiment-largest-molecules.html
 - Furuta, Aya (2012), “One Thing Is Certain: Heisenberg’s Uncertainty Principle Is Not Dead”, Scientific American
 - http://www.azonano.com/article.aspx?ArticleID=1725
 - http://www.nature.com/news/physicists-snatch-a-peep-into-quantum-paradox-1.13899
 - A. Voje, J. M. Kinaret, and A. Isacsson,“Generating macroscopic superposition states in nanomechanical graphene resonators”, Phys. Rev. B 85 (2012)
 - http://phys.org/news/2014-01-quantum-mechanics-efficiency-photosynthesis.html
 - Baglione, Melody L. (2007). Development of System Analysis Methodologies and Tools for Modeling and Optimizing Vehicle System Efficiency (Ph.D.). University of Michigan. pp. 52–54.
 - http://www.qrg.northwestern.edu/projects/vss/docs/power/2-how-efficient-are-solar-panels.html
 - http://phys.org/news/2015-11-quantum-superposition-events.html
 - http://www.flownet.com/ron/QM.pdf