A quantum field theory is modeled by a probability distribution (measure) over the space of all field configurations, specified implicitly by an action functional. We seek to describe systems by characterizing their distributions using moments (correlation functions).

A simple characterization for the multi-variate distribution (each point corresponds to one random variable) is to specify it's mean at every point in space. The mean field approximation does exactly that -- it neglects all "fluctuations" in field values at each point and considers a classical "field profile". Commonly, this field profile is also assumed to be uniform in space, so that one may conveniently solve for self-consistent answers for the background field value. Also, note that what is usually computed in physics (lowest action configuration i.e. max-likelihood) is actually the "mode" of the distribution -- but the mode and the mean are interchangeable if the distribution is peaked and large fluctuations have negligible measure. (Note: If the system is not in that regime, then these approximations/truncations are useless anyways, so splitting hairs regarding the terminology here is quite pointless)

Quantum mechanically, the true solution is the superposition of a bunch of configurations which are modeled as "fluctuations" around the "mean" field. Neglecting any interactions between these fluctuations at different points (suppressed by a coupling constant) the leading order dynamics is captured by a quadratic Lagrangian — just the kinetic/gradient terms for the fluctuations. As this action leads to a Gaussian measure on the fluctuating degrees of freedom, it is also referred to as the Gaussian approximation. This is equivalent to treating each fluctuation degree of freedom as an independent harmonic oscillator (essentially "free" field theory).

Whether each of these approximations is useful depends on details like the size of the coupling constant, dimensionality of the system, etc.