What is meant by the term "Fourier mode"? In physics, whenever Fourier analysis is utilised to analyse a problem the term "Fourier mode" is often used, e.g. "a given function can be represented in terms of its Fourier modes".
My question is, what exactly is meant by the term "Fourier mode"?
Is it in reference to a given wave oscillating at a fixed frequency? And then a given function is built up from an (infinite) superposition of these waves, each oscillating at a fixed frequency (ranging over a continuous frequency spectrum)?!
 A: A mode in physics is, generally speaking, the spatial part of a waveform. A Fourier mode, more specifically, is a wave that oscillates sinusoidally in space. Thus, when we write the Fourier transform of a wave function $f(\mathbf r,t)$ as
$$
f(\mathbf r,t) = \int\tilde f(\mathbf k,t)e^{i\mathbf k\cdot \mathbf r} \: \mathrm d\mathbf k,
$$
the quote you give,

a given function can be represented terms of its Fourier modes,

means exactly that: we have a bunch of Fourier modes, which are the functions $ f(\mathbf k,t)e^{i\mathbf k\cdot \mathbf r}$, with $\mathbf k$ a static parameter, and then we add them all up (via integration over this parameter) to give us back our $f(\mathbf r,t)$.
The term 'mode' isn't used by itself very much, but it does appear with other qualifiers such as the normal modes of a system, which specifically requires a regular sinusoidal oscillation at each and every point of the system. Similarly to Fourier modes, these normal modes can then be used to represent any arbitrary wave as a sum of normal modes.
It's also important to note that 'mode' in general can apply to both continuous and discrete systems, as does 'normal mode'. You can in principle define Fourier modes for an infinite or cyclical chain of discrete coupled masses, characterized by a sinusoidal dependence on the (discrete) spatial index, but this is rarely used. In all of these cases, though, what sets the Fourier modes apart from other modes of oscillation is the sinusoidal spatial dependence.
