Because it is a solution to the time-independent Schrodinger equation
and it satisfies the boundary condition of your problem,
this solution describes a state of specified energy, in the sense that the fluctuation of the energy $\Delta E_n=0$. This is easy to show since
\Delta E^2&= \langle \Psi_n\vert H^2 \vert\Psi_n\rangle -
\langle \Psi_n\vert H \vert\Psi_n\rangle^2\, , \\
(If there is no fluctuation in energy you can assign a definite energy to this solution.)
Because only certain values of $E$ can lead to solutions of (1) that satisfy the boundary conditions, and because these values can be labelled by $n$, the various $E_n$ values thus give you a sequence of solutions for which the energy $E_n$ associated with this solution does not fluctuate: in other words, you can speak of solution $\psi_n(x)$ "having definite energy $E_n$". This is quite useful because energy is conserved, so speaking of a solution $\psi_n(x)$ which has definite energy is something that remains true in time.
In this way the value of $E_n$ functions a time-independent label, and you can "name" solutions using energy, i.e. you can say that your system is in the state having energy $E_n$. Given $E_n$ is then enough to completely identify the state $\Psi_n(x)$ and use $\Psi_n(x)$ for whatever purpose you need. (In higher dimensional problems you need to specify more than just $E_n$ to uniquely identify the state, but in 1d this is enough.)
Superpositions are solutions which satisfy the time-dependent Schrodinger equation, and in particular satisfy the boundary condition of the problem. They are legitimate solutions but do NOT have definite energy, i.e. for those solutions $\Delta E\ne 0$, in constradistinction with the states $\Psi_n(x)$ which are solution to the time-independent Schrodinger equation. In general, there is nothing to suggest that your system will be in a state of definite energy (in the sense that there is no reason to suggest $\Delta E=0$) and it will be a superposition of $\Psi_n(x)$. The precise superposition depends on the initial condition of your problem, i.e. on the initial wavefunction at $t=0$, much like a string clamped at both ends that has some weird initial shape.