Difference between single mode and multi mode optical fibres? What is the difference between single mode and multi mode optical fibres? First off, I guess that by modes we mean the spatial modes of the electric (or magnetic?) field right?
Now: what makes a fibre able to support more than a single mode? I mean, what aspect of its structure corresponds to which mode(s) can be transmitted?
 A: It's simply the diameter of the fiber core. In a single-mode fiber, only the lowest-order mode fits physically into the fiber.
A: You can assume either electric or magnetic field for simplicity, they both are present in light as it's electromagnetic wave.
 
Now, mode is a sustainable pattern in the fiber optic cable. Imagine waves on a string, only the integral multiples of half wavelengths effectively form a mode. 
Let there be a complex output pattern at the output of a string. It turns out if we aren't playing with it continuously, it will be essentially periodic, maybe long period.
And we can always reduce it via fourier transforms to a sum of some basic modes. 
Imagine the output at the end of a MMF(multi mode fiber) as waves on a string with intensity output as the wave amplitude on the string. It may be complex. But you can simplify the output by considering it as a sum of several modes.
You can play with waves on a string here. It won't do a fourier transform though.
http://phet.colorado.edu/sims/wave-on-a-string/wave-on-a-string_en.html
The concept is a little different for SMF and MMF.
In case of SMF, Ray theory fails and light propagates as a wavefront as explained here:
Single-mode fibers and ray-theory of light
A: The V-number defines the mode of any fiber. It depends on wavelength, diameter of fiber and refractive index.

The v number is
\begin{align}
\ V^2 &= \frac{2}{} (_1^2−_2^2)\\\\
 \end{align}
Each mode has an effective index that can be defined by:
\begin{align}
\ _ = _ \frac{2}{}\\\\
 \end{align}
The effective index tells you how tightly the mode is confined to the waveguide core

*

*Guided mode _ < _ < _


*Tightly confined to the core _ ~ _


*Weakly confined to the core _  ~ _


*Unguided or radiating modes _<_
