boundary conditions
Boundary conditions are the physical demands you place on a wavefunction at the edges of a problem, and they are what single out which solutions nature actually permits. The Schrödinger equation by itself has far too many mathematical solutions; most of them are physical nonsense. Boundary conditions are the filter that throws out the nonsense and keeps the real states.
The usual requirements are common sense dressed in mathematics. The wavefunction must not blow up to infinity far away, since a particle has to be found somewhere with finite total probability. It must vanish where the walls are impenetrably high. And wherever the potential is finite, it must join up smoothly, with no breaks or kinks. These conditions are rarely satisfiable for arbitrary energies.
And there lies the deep payoff: insisting on good boundary behaviour is exactly what forces energy to be quantised. Only at certain special energies does a solution manage to stay finite, vanish where it must, and knit together smoothly all at once. Those surviving energies are the eigenvalues; the discreteness of atomic energy levels is the boundary conditions speaking.
Demanding ψ vanish at both walls picks out a discrete ladder of allowed energies.
Boundary conditions depend on the potential. Where the potential jumps to infinity, ψ must vanish but its slope may kink; where the potential is merely finite, both ψ and its slope must stay continuous. Using the wrong condition gives wrong energies.