Updated: Apr 20
Quantum mechanics, is a branch of science dealing with the behaviour of matter and light on the atomic and subatomic scale. It attempts to describe and account for the properties of molecules and atoms and their constituents—electrons, protons, neutrons, and other more esoteric particles such as quarks and gluons.
I'll attempt to explain quantum mechanics to the best of my ability. Before attempting to learn anything about quantum mechanics, though, you should understand where the quantum realm lies and why it is separate from the classical realm. The best comparison, I think, is that of the Earth.
If you look around and attempt to measure several meters across the ground, from your perspective, the Earth is flat. If you go into space and look at the Earth from there, though, you will be able to see that the Earth is most definitely round, you just happen to live in the flat Earth realm which is the limiting case of the round Earth realm.
In a similar manner, classical mechanics is what you observe in the limiting case of low speed and large size; at very fast speeds, the effects of relativity start becoming noticeable and at very small sizes, the effects of quantum mechanics start becoming noticeable.
In physics, quantum mechanics refers to a few basic postulates governing the quantum world and the consequences of these postulates; since this answer is intended for a general audience, I will focus on a few of the effects resulting from these postulates rather than the postulates themselves which I don't think will be substantially interesting or enlightening for non-physicists.
So what are these effects? One of the most important is wave-particle duality.
According to wave-particle duality, every particle is a wave and every wave is a particle. What this means is that particles possess wave-like properties (they have phase, they have a frequency, and they can interfere with other particles) and waves possess particle-like properties (photons, the particles associated with electromagnetic waves, for example, can be absorbed and emitted discretely just like particles).
At large sizes, the wavelength of an object will be negligible relative to the size of the object, just as the curvature of the Earth becomes negligible as you zoom on really close to it. Another important feature of quantum mechanics is the uncertainty principle which states that the product of the uncertainty of a particle's position and momentum must be greater than a certain value, in other words, the position and momentum of a particle cannot both be precisely known. This has to do with the probabilistic nature of quantum mechanics.
When you measure a particle's momentum or position, there are multiple values that you might measure with different probabilities. After many measurements, you will arrive at a distribution averaged around what you would expect to measure classically with some standard deviation. Per the uncertainty principle, the smaller the standard deviation in position is (i.e. the higher the chance that you will measure your particle near the classical value), the greater it will be in momentum and vice versa.
At large scales, the magnitude of uncertainty is so small that it won't even come close to the precision of the instrument being used to take the measurements. Interesting to note is that the uncertainty principle is an inherent property of wave-like systems and is not unique to quantum mechanics.
Since any given particle or system of particles is a wave, it can be represented by a wave function. This wave function describes the state of the particle/system and can be used to determine properties of the system such as position and momentum. Of course, in accordance with the uncertainty principle, this wave function won't give you the exact momentum and position of a particle but rather the probability that the position/momentum is within a certain interval.
One property of waves not yet mentioned is that they have boundary conditions. If you create a wave on a string that is fastened at both ends, for instance, then the equation for that wave must be zero at both ends of the string. This feature leads to quantization which means that the system can only exist in a discrete (rather than continuous) set of states satisfying the boundary conditions. In the case of the string, the wave can exist in any of an infinite number of discrete states- a state with no nodes, a state with one node, a state with two nodes, etc.
Quantum Mechanics is a theoretical discipline intended for systems so small that renders classical mechanics inapplicable.
By inapplicable, I mean that all matter has a wavelike behavior associated with it and in systems that are small enough (atomic/subatomic dimensions) this wave nature becomes quite significant.
So when someone asks you “where is a wave?”, you'd be quite confused because a wave is spread out over a region. This immediately leads to the uncertainty principle that : The more precisely we know the speed of a quantum object, the less precisely can we measure it's position. Even though in our common sense this appears quite nonsense but we must bear in mind that this “common” sense we have developed is due to the fact that we are exposed to objects with orders of magnitudes larger than atomic dimensions by many orders if magnitude. The so called common sense of the macroscopic world doesn't work in the atomic realm.
In order to work out systems of such small dimensions, and respecting the uncertainty principle the theory of quantum mechanics is developed wherein the concept of exact position or momentum of a particle is not considered, rather the probabilities of finding positions or momentums in a given range is considered. In that, the theory is probabilistic rather than deterministic.
Don't however feel that quantum mechanics has got something wrong or incomplete because it only talks about probabilities, it has never been shown to fail. Even though there have been thought experiments showing how quantum interpretations fail in the classical world, they work completely fine in the quantum domain. Also, calculations based on quantum mechanics (ex - hydrogen spectrum) have been found to be extremely accurate.
However, even though quantum mechanics is extensively used in calculations, it is not well understood. Infact, how does the quantum description of the world lead to the reality we perceive is still an open problem to work on which started right from the famous Bohr-Einstein debates.