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.
