What does the quantum state of a system tell us about itself? 
In quantum mechanics, quantum state refers to the state of a quantum
  system. A quantum state is given as a vector in a vector space, called
  the state vector. The state vector theoretically contains statistical
  information about the quantum system.

The article in Wikipedia about quantum state suggests that the quantum state of a system is indeed its state vector which contains statistical information about it.
But what are those statistical information? The particle's location, momentum, wavefunction and.. energy level? 
When I measure the momentum and position of an electron, the quantum superposition principle suggests that I'm only getting a result corresponding to one of its possible state because of it exhibits both wave and partile behaviors. But what exactly is a possible state? 
 A: Although the concept of state can be well defined, at some level it takes a certain level of abstraction to really understand what a state is.  From a conceptual point of view, it is easier to think of a state in a classical context.  In a classical context a state is simply a particular configuration of objects that are used to describe a system.  For instance, in the case of a light switch we can talk about it being in an on or off state (e.g. the light switch can be in the "on state" or the "off state").  In quantum mechanics this situation is a little more complicated, because we add a level of abstraction that allows us to consider the possibility of the superposed states where our knowledge of the switch is insufficient and we must consider it to be in an "on and off" state.  However, this state is not a classical state in the sense that we could ever observe the switch in the "on and off" state, it is a quantum state that exists in an abstract space called Hilbert space.
Every state of a system is represent by a ray (or vector) in Hilbert space.  Hilbert space is probably most simply understood by creating a basis that spans the space (e.g. that is sufficient to describe every point in the space) as a long summation of complex variables, which represent independent functions.  Any state, or ray in the Hilbert space, can then be understood using Dirac's bra - ket notation.
The ket is more commonly used and a state is represented as $|\psi\rangle$. It is important to understand that the symbol inside the ket ($\psi$) is an arbitrary label, although there are commonly accepted labels that are used throughout physics, in general the label can be anything a person wants it to be.  
In the case of considering the a state be projected onto some basis, we can write this mathematically as: $$|\psi\rangle = \sum_i |i\rangle\langle i|\psi\rangle$$   In this representation the $\langle i|\psi\rangle$ takes on the role of a set of complex coefficients $c_i$where $|i\rangle$ serves to represent the each of the $i$ basis states.
In the early development of quantum mechanics, the question of describing atoms and predicting their properties was the main goal.  Many of the questions physicists were interested in centered around questions of energy, position and momentum transitions. Because of this fact, most of quantum descriptions of reality are centered around finding a means of representing energy and momentum states of particles, particularly electrons, surrounding the nucleus.  The quantum mechanical description of electrons surrounding an atom is therefore focused on describing the probabilities of finding an electron in a particular orbital state surrounding the atom.  The state vector is thus used to represent a ray in Hilbert space that encodes the probability amplitude (essentially the square root of a probability, which is understood to be a complex number) of finding an electron in a particular orbital state (e.g. position, momentum, spin). 
This is an example of applying quantum mechanics to help resolve a particular physical problem. I make this distinction, because quantum mechanics is simply a means to an end, and thus must be understood as a tool to be used to describe a particular physical situation and to predict certain physical outcomes as the system evolves.  One of the core debates of the 20th century centered around whether quantum mechanics could provide a complete description of the universe. The answer to this question is yes, and has been affirmed in repeated experiments.  
A: It tells you the probability distributions of every measurable you can perform on the system (momentum, energy etc), the likelihood of every outcome. The wavefunction contains everything there is to know about your system. If you have 10000 identically prepared systems, then you will gain no new information about system 10000 if you perform any type of measurements on systems 1 to 9999.
A possible state is a normalizable solution to the Schrodinger equation. After a measurement the system will be in an eigenstate of the operator corresponing to your observable.
A: Actually the state of a quantum system is not an accurate representation of what it is "internally". That is it is not a ontic state in that it captures everything about the system. It is only an epistemic state in that if we perform a measurement of an observable, then what are the real world values of it ( think spin up and spin down), the state is decomposed in terms of these states. So we can think of the $|\Psi\rangle$ as an observer's magnifying glass rather than the 'real thing'. So the quantum world reveals its interactions with us through this state variable. However the real thing is something which has evaded us !!!. this is why Einstein was perturbed and asked "Do you really believe that the Moon does not exist if we don't look(observation) at it?"
Einstein believed that it is the fault of the description and not the system itself, while QM has shown that it is not so. The system itself doesn't have an ontic state.
