What is a quantum state? I've read wiki definition and also some other questions here on StackExchange, but I still can't quite catch what is a quantum state, what are properties of said state. I also was looking for some basic example of it but I couldn't find any. I would appreciate an explanation for a true beginner.
 A: State in general
Generally speaking, a state of a system is some mathematical object that completely describes the system at a particular time. For example, the state of a free classical particle is identified by its position $x$ and momentum $p$ and the state of a computer is identified by the values stored in all its registers and memory. In many theories, there are two fundamental calculations that involve a state. First, we can use the laws that govern state evolution to compute the state at time $t_2$ based on the state at time $t_1$. Second, we can use a state to predict the outcomes of observations.
This general picture applies to classical mechanics, thermodynamics, chemistry, control theory, software engineering, quantum mechanics etc.
State in quantum mechanics
Perhaps the best way to intuit what the quantum state is is to consider how it connects to observable quantities. In quantum mechanics, measurement is probabilistic and the probability distribution over outcomes of a measurement is constructed from two ingredients:

*

*an observable $A$, which is a mathematical object describing the measurement to be performed,

*a quantum state $\rho$, which is a mathematical object describing the quantum system at a given time$^1$.

Given these two ingredients one can compute the probability $p(\lambda|\rho, A)$ that the measurement of $A$ on a system in state $\rho$ yields the outcome $\lambda$. The recipe for computing $p(\lambda|\rho, A)$ is known as the Born rule.
Quantum theory also tells us how quantum states evolve, so we can find the state $\rho_2$ of our system at time $t_2$ from its state $\rho_1$ at time $t_1$. See Schrödinger equation for details.
Example
The simplest quantum mechanical system is a qubit. Observables on a qubit are modeled as complex $2\times 2$ Hermitian matrices and states are modeled as complex $2\times 2$ positive semidefinite matrices with unit trace. Thus, the following matrix is an example state of a qubit
$$
\rho = \frac12\begin{bmatrix}1&1\\1&1\end{bmatrix} = |+\rangle\langle +|.
$$
We can also consider an example observable
$$
A = \begin{bmatrix}1&0\\0&-1\end{bmatrix} = |0\rangle\langle 0|-|1\rangle\langle 1|
$$
to see how $\rho$ determines probabilities of the outcomes of a measurement of $A$. The eigenvalues of $A$ are $1$ and $-1$ and so the only possible outcomes of a measurement of $A$ are $1$ and $-1$. The Born rule says that
$$
p(\lambda|\rho,A) = \mathrm{tr}(\rho P_\lambda)\tag1
$$
where $P_\lambda$ is the projector onto the eigenspace of $A$ associated with $\lambda$. Let's use $(1)$ to compute the probability that our measurement of $A$ on a qubit in state $\rho$ yields $-1$. Noting that $P_{-1} = \begin{bmatrix}0&0\\0&1\end{bmatrix}$, we have
$$
p(-1|\rho,A) = \mathrm{tr}\left( \frac12 \begin{bmatrix} 1&1\\1& 1\end{bmatrix} \cdot \begin{bmatrix}0&0\\0&1\end{bmatrix} \right) = \frac12.
$$
This exemplifies how for each observable $A$ our state $\rho$ determines the probability distribution over outcomes.

$^1$ We use the Schrödinger picture here.
A: There's a very good discussion of this in a paper with almost the same title: "Peres, Asher. "What is a state vector?." American Journal of Physics 52.7 (1984): 644-650."
Peres argues that a state vector (or quantum state) represents a procedure.  In Sec. III, Peres quotes "Giles, R., 1970. Foundations for quantum mechanics. Journal of Mathematical Physics, 11(7), pp.2139-2160"

"We define a state of a system to be a (nonempty) collection of
methods of preparation of the system, where by a method of preparation we mean a document giving detailed instructions for the
preparation".

In other words, writing $\vert\psi\rangle$ should be understood as meaning that some manipulations have been made and we give an interpretation of result of these manipulations as the quantum state $\vert \psi\rangle$.
The discussion of Peres is quite approachable, and he is careful to give examples of contradictions with various other interpretations of a state.  There's a follow up discussion on the time evolution of quantum states.
A: I think the best way to describe quantum states, is by realizing that they are just a mathematical way of representing real structures or phenomena. The real interpretation of the quantum wave function is still an open question, however, the one thing we do know is, different structures are represented through different states.
Imagine an electron in the hydrogen atom. Forget all the advanced quantum mechanics for a second. Imagine, the electrons are particles revolving in $K,L,M,N...$ shells. From Bohr's theory, you can easily check that electrons in different shells have different energies. The point is, you can have an hydrogen atom in a ground state, where the electron is in $K$ shell. However, you can also have an atom in an excited state $M$ shell, for example. Even though these are both hydrogen atoms, they are different. The quantum wavefunction or 'state' is a mathematical way of representing these different structures, so that you can easily differentiate between them.
Similarly, there are other examples of this. What I'm getting to, is that a quantum 'state' does not have different properties as such. It is a 'collection' of all the properties of the system you are describing. They are a way of mathematically representing a system, like the different hydrogen atoms in the previous example. It is a mathematical description of a system, that allows you to distinguish or compare it to another system.
