The key to understanding this issue is that $\lambda$ is not a fixed parameter, so $-\phi''+\lambda\phi=0$ for example isn't just a single equation: it's an entire family of them. There are initial or boundary conditions imposed, in which case there are infinitely many $\lambda$ for which this form of differential equation has a solution. For any such eigenvalue $\lambda$, the solution is unique (up to rescaling), so there is exactly one eigenvalue associated to each eigenfunction. In the context of quantum mechanics this means there are an infinite number of allowable energy levels. Furthermore, the fact that the spectrum, or set of eigenvalues, of the operator is discrete (countably infinite with no limit points IIRC) means that the energy levels are quantized, as we should expect in quantum mechanics.