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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?

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It's simply the diameter of the fiber core. In a single-mode fiber, only the lowest-order mode fits physically into the fiber.

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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

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The V-number defines the mode of any fiber. It depends on wavelength, diameter of fiber and refractive index.

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The v number is

\begin{align} \ V^2 &= \frac{2𝜋𝑑}{𝜆} (𝑛_1^2−𝑛_2^2)\\\\ \end{align}

Each mode has an effective index that can be defined by: \begin{align} \ 𝑘_𝑧 = 𝑛_𝑒𝑓𝑓 \frac{2𝜋}{𝜆}\\\\ \end{align}

The effective index tells you how tightly the mode is confined to the waveguide core

  • Guided mode 𝑛_𝑐𝑙𝑎𝑑 < 𝑛_𝑒𝑓𝑓 < 𝑛_𝑐𝑜𝑟𝑒

  • Tightly confined to the core 𝑛_𝑒𝑓𝑓 ~ 𝑛_𝑐𝑜𝑟𝑒

  • Weakly confined to the core 𝑛_𝑒𝑓𝑓 ~ 𝑛_𝑐𝑙𝑎𝑑

  • Unguided or radiating modes 𝑛_𝑒𝑓𝑓<𝑛_𝑐𝑙𝑎𝑑

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