Mathematically speaking, modes (in optical fiber or waveguide) are eigensolutions of the wave equation, and they have a certain polarization (in the general case they are hybrid modes). In a fiber, you cannot really describe a mode in a 'ray-picture', things are more complicated because you have to use cylindrical coordinates, there are no planar interfaces. The number of modes you get in a fiber depends on the geometrical dimension of the core. In practice, when you couple in the light from an external source, it can go into any of the available modes.
Hope this explanation helps.