= Solution
$V(x)$ is the particle's <potential energy> as a function of position, and $E$ is its total energy <eigenvalue> in this <time-independent Schrödinger equation>. Solving it with the boundary conditions determines the allowed <stationary states> and energy levels.
The <wavefunction> is a <probability> amplitude: the <Born rule> gives position <probability> <density> $|\psi(x)|^2$, normalized to unit <integral> for one particle. $\psi$ itself need not be real or positive and is not a classical matter-density profile. The full stationary-state time dependence is $\Psi(x,t)=\psi(x)e^{-iEt/\hbar}$, leaving $|\Psi|^2$ time independent.
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