In some situations, you can have continuous spectra, such as the energy bands in crystals.
Eigenvalues are easiest to understand in terms of linear algebra. A square matrix represents a transformation on some vector space; the eigenvectors are the directions in which the matrix acts solely as a scaling transformation, and the eigenvalues are the corresponding scale factors.
I find it easiest to understand that first, and then look at Quantum Mechanics as a generalization of linear algebra to an infinite-dimensional Hilbert space. An operator in Hilbert space is in some sense an infinite-dimensional square matrix, and the eigenstates are infinite-dimensional column vectors.
Another way to think of it physically is in terms of resonance frequency and normal modes. Take a violin, for example. It has a fairly complicated shape. But for some frequencies of vibration, only one mode of oscillation is excited. This mode is an eigenstate; the frequency (or its square) is an eigenvalue.
The modes of vibration of some arbitrary shape can be mapped out by putting a thin layer of sand on the surface, and using a speaker-like device to drive vibrations at a particular frequency. At resonance frequencies, the sand forms a stationary pattern of gaps corresponding to the nodes of the standing wave pattern (or was it antinodes? I forget). The pattern you see is an eigenstate of that shape; the (square of the) frequency is its eigenvalue.