Classical and quantum systems What are the main differences between a quantum and classical system? How does one can distinguish them?
 A: 
What are the main differences between a quantum and classical system? How does one can distinguish them?

An experimentalist's answer, in other words, why do we need two different theories.
In a classical system if we do experiments with massive bodies , they have in principle well measured (x,y,z) positions at time t. From falling apples to the planetary system an elegant mathematically theory was developed called classical mechanics, modeling our everyday world .
In studying the wave properties of light the theory of classical electrodynamics was developed, complementing classical mechanics describing and predicting new experimental situations. 
With thermodynamics and statistical mechanics the tool box of theories for describing physical reality was considered complete. There even were statements of prominent physicists of the nineteenth century that now only engineers would be needed, physics was complete.
Then the worm turned, because experimental observations appeared that were not predicted and could not be theoretically explained within the classical system, and the mathematical theory of quantum mechanics had to be invented.
What were these observations that separate a classical from a quantum mechanical system?
1) Black body radiation, which could not be explained classically and had to assume that the electromagnetic radiation came in quanta, units, carrying energy h*nu
2)The photoelectric effect which also needed quanta ( packages of energy) for the electrons emitted from materials
3)The light spectra from atoms, both emission and absorption, which instead of showing the classical continuous behavior showed absorption and emission lines, again showing discrete packages, quanta , of energy . For hydrogen these lines even followed mathematical series.
Trying to understand a model of how electrons could revolve around nuclei , incompatibility with the classical EM  theory was found .  There was no reason that the electrons would always stay in fixed orbits around the nucleus. Fixed orbit solutions with classical electromagnetism could be found, but there was no theoretical reason once disturbed that the electron would not fall into the nucleus diminishing its charge by one. 
The constraint was imposed by Bohr, that the orbits had to be fixed/quantized with the Bohr atom. It could explain the Balmer series, but it was an ad hoc model, not a theory. The time came for the Schrodinger  equation to enter the stage which allowed the development of a mathematical theoretical model that explained not only simple atoms , but set the stage for the further theoretical study of elementary particle interactions. It was complemented with the Born rule, which is the postulate that allows to relate theoretical predicted numbers to data
We are now at the stage where just these two frameworks , the classical and the quantum mechanical , are sufficient to explain observations and predict new phenomena successfully, mainly for large dimensions classically, and for small dimension quantum mechanically. The real size is determined , as stated in other answers, by the value of h_bar and the mathematical relations that constrain the solutions, which give rise to the Heisenberg uncertainty principle. Sometimes, as in superconductivity, quantum mechanical effects can exist in large dimensions, but in general the framework is the underlying one for small dimensions. As dimensions grow large the quantum mechanical mathematical formulations lead at the limit consistently to the classical formulations.
A: The main difference between classical and quantum physics is the fact that observables (Hermitian operators whose eigenvalues determine the possible values of physical quantities of the system) do not commute. For classical systems commutativity is trivially satisfied. 
A direct consequence of this is the uncertainty principle and the fact that via Bell's theorem there is no local realism. 
If it wasn't for this simple but important fact, quantum mechanics could essentially be reduced to a classical (albeit stochastic) theory. 
A: $\newcommand{\pdv}[2]{\frac{\partial #1}{\partial #2}}$
$\newcommand{\ket}[1]{\left| #1 \right>}$
Quanta vs. Continuous
Quantum, as the name suggests, deals with quanta ie packages of something. For example for a given particle and given potential the energy is quantised. Eg. for quantum mechanical SHO the energy levels are $E_n=\hbar \omega (n+\frac{1}{2})$. Notice that the only allowed energies in this system are $E_0= \hbar \omega \frac{1}{2}$, $E_1= \hbar \omega \frac{3}{2}$, $E_2= \hbar \omega \frac{5}{2}$ etc. However there shouldn't be any confusion that QM forbids energies like $E=\hbar \omega \frac{1}{4}$ or any energy for that matter. For this specific potential and for a given $\omega$ however this is not a possible energy eigenvalue.
Position and Momentum vs. Wave Function
In classical mechanics a system is completely described by a point on the phase space, which usually consists of the momentums and positions of the objects involved. In QM an isolated system of one or more particles is completely characterised by a complex-valued function called the wave function and usually denoted with greek capital letter psi $\Psi$ or with lower case psi $\psi$. Note that the information about the position and the momentum of the particle is “encoded” in this wave function. Some times the system is described by a vector $\ket \psi$ called ket in a Hilbert space, which is of course mathematically equivalent to the wave function.
Deterministic vs. Probabilistic
Classical mechanics describes a deterministic system, which means that what will happen at the next instant of time is completely determined, where as QM describes a probabilistic system. The norm squared of the wave function $|\psi|^2$ gives the probability density. Notice that in QM a wave function can be in superposition of more than one states ie. $\psi=\alpha \psi_1+\beta \psi_2$ then the probability that the particle is in state $\psi_1$ is given by $|\alpha|^2$ and the probability that the particle is in state $\psi_2$ is given by $|\beta|^2$. Since the total probability must add to one the wave function is usually normalized in such a way that $|\alpha|^2+|\beta|^2=1$.
Mathematical Differences
There is also mathematical difference between them. CM almost always deals with real numbers. You can see here and there some complex numbers but they are mainly computational tools, which have no physical meaning. On the other hand QM is governed by imaginary numbers, which have actual physical meanings. In particular the wave function is a complex valued function, which can be interpreted as probability amplitude.
Equations of Motions
The equation of motion for classical physics is a nice second order ordinary differential equation known as Newtons Law:
$$\vec F = m \ddot a$$
Sometimes you also see it in the following form:
$$m \ddot a = - \nabla V$$
The equation of motion for QM is a second order in space and first order in time partial differential equation known as the Schrödinger equation:
$$i\hbar\pdv{}{t}\Psi(\vec r,t)=\frac{\hat { \mathbf {p}}^2}{2m}\Psi(\vec r, t)+V(\vec r)\Psi(\vec r, t)=-\frac{\hbar^2}{2m} \nabla^2 \Psi(\vec r,t)+V(\vec r) \Psi(\vec r,t)$$
or in matrix form as:
$$i\hbar\pdv{}{t}\ket{\Psi(\vec r,t)}=H\ket{\Psi(\vec r,t)}$$
Orders of Magnitude
The QM can be considered as the real world since it describes the world of atoms and what you see around you is made of atoms. The classical mechanics is usually an approximation of QM. For example you can take the mass of the particle to be large or the energy level of the system to be large and you usually arrive at classical predictions. Notice that QM deals with energies of the order $10^{-34}$ Joule and masses of the order $10^{-23}$ kg. This should give you a feeling of how small the scales are.
In general you can make the assumption that if you are dealing with particles like electrons, protons photons or whatever you should consider the system to be a quantum mechanical system. However, in most applications quantum objects can be considered as classical object for convenience.
There are probably more differences but these are the major ones that come to my mind.
A: In a classical system (as we have studied), the behavior of a body can be exactly determined. For example, in projectile motion, the position, velocity, and acceleration of a system can be determined exactly by knowing only the time $t$ plus your initial conditions. In terms of size a classical system usually deals with macroscopic systems.
In a quantum system, you are bound to uncertainties, and therefore, to a range of values in determining the motion of the body, as stated by Heisenberg Uncertainty Principle. In addition, we are dealing with particles and Waves, and the energies that these waves exhibit.
But quantum systems can also be treated classically. For example, in geometric Optics, we focus on the path that light travels. However, if we focus on the particles of light that were in motion, we are dealing with quantum systems of an ensemble of particles.  
