There is of course a purely mathematical answer to this question (the Wikipedia pages can contain it), but it's not the only one.
As regards (metallic, dielectric, ...) waveguides: a
mode is a field configuration which
- is a solution of Maxwell's equations and satisfies the boundary conditions of the waveguide;
- is able to keep a uniform magnitude along the direction of propagation;
- is self-consistent along the guide (that is: a field which behaves as if it were a plane wave in free space, but along the direction of propagation inside the waveguide). This implies that, after two consecutive reflections on the waveguide boundary, the field is able to be the same as before the reflections.
This is why it is a "stable" solution.
It is the structure of the waveguide, together with the boundary conditions, that determines what field configurations are allowed to meet the above features: for this reason, the modes are often considered as the
eigenfunctions of the structure, with the related propagation constants being the
eigenvalues. They represent the kind of waves that the waveguide structure can
naturally host.
Maybe the main difference between a waveguide mode and the standing waves of a string is
propagation: a mode is a field which
travels along the waveguide axis; a standing wave is, by definition, not able to propagate. But yes, they can both be described by a fixed field structure.
IIRC, in the case of optical fibers, each mode has a related
angle of incidence and it can be depicted as a ray, so yes, I think it is often correct to consider the number of rays equal to the number of currently active modes.
http://www.eecs.ucf.edu/~tomwu/course/eel6482/notes/19%20Parallel%20Plate%20and%20Rectangular%20Waveguides.pdf (page 3) is one of the simplest kind of modes you can obtain:
$$\mathbf{E}(z) = - \displaystyle \frac{V_0}{d} e^{-j k z} \mathbf{a}_y\\
\mathbf{H}(z) = \displaystyle \frac{V_0}{\eta d} e^{-j k z} \mathbf{a}_x$$
This is a field which is constant along ##y##. The direction of propagation is ##z## and the magnitude of the Electric field ##V_0 / d## is constant for
all ##z## (the magnitude of the Magnetic field is constant, too). It is also a TEM mode, because both the Electric and Magnetic fields are orthogonal to the direction of propagation.
