How should I think about the word "coupling" in quantum field theory? People doing any form of quantum field theory (QCD, string theory etc) always use the word "couple" and I am not sure exactly what it means. If a QFT couple to gravity I can make an educated guess that it means that includes gravity in some sense. But I only have a vague intuition.
Is the fact that something is coupled to something else mathematically well-defined? Do people use the term differently or does it always mean the same?
It is one of these words I have heard a thousand times but never really understood. '
Any insight is very appreciated.
 A: 
Is the fact that something is coupled to something else mathematically well-defined?

Sort of, because when you say that X and Y are coupled to each other, it means that X and Y interact with each other. This interaction is encoded in a mathematical way, and can appear in many places in different ways (in the action/Lagrangian, in the equations of motion, in the physical scattering amplitude, in the gravitational potential, etc.). I say 'sort of' because the exact mathematical expression which encapsulates the occurrence of the coupling depends on the quantity you're interested in studying. The fact that X and Y interact with each other is a physical statement, which has physical implications - a universe without interactions among any entities is boring because nothing is created, destroyed or modified. Fortunately, our universe is not like that, and hence we exist. X and Y here can be anything, but usually, they correspond to 2 different fields. One counterexample is when people say that the Ricci scalar $R$ (not a field) is coupled to a scalar field $\phi$ in a scalar-tensor theory of gravity.

Do people use the term differently or does it always mean the same?

They typically mean that two entities X and Y, whatever they might be, interact with each other. What X and Y are, and what the mathematical expression for their interaction is, is usually clear from the context.
Usually, in a QFT, you need to perform perturbation theory to compute physical observables (scattering amplitudes). Amplitudes depend on the coupling $g$. Perturbation theory just means that you approximate the exact answer by writing only the first few terms of it by performing a series expansion in integer powers of $g$. Sometimes you need non-pertubative effects, and even there you will get a functional relationship of the quantity you're interested in in terms of g, e.g. the Schwinger electron-positron pair production rate depends (schematically) on $e^{-1/g}$ where g = electric charge (coupling).
In gravity (GR), the coupling is $\sqrt{G}$, which is quite small (in SI units, for instance), which is another way of saying that one can approximate GR as a weakly-coupled QFT of gravitons (as long as the energy scales probed are smaller than the Planck scale). This small coupling then gives the physical result that 2 electrons will repel each other due to electrostatic forces much more than they will attract each other due to gravity.
In short, a coupling is just a parameter, which tells you how much/how quickly/how slowly something is created/destroyed/modified/exchanges energy and momentum with its surroundings.
A: Typically it is well defined. In a given Lagrangian you can have coupled terms (let's say two fields), which means that you multiply those two fields and the strength of their interaction is measured by the "coupling" constant multiplying the product. It is essentially a number, that, even though a constant, may depend on the energy scale (used in RG flow methods). This is the standard coupling.
There is something called "minimal coupling", which means, that there is no exact term that multiplies two fields, but  the two (or more) fields are connected via the Lagrangian. An example: the quantum gravity model called Causal Dynamical Triangulations uses a discretization called Regge calculus (Loop quantum gravity also use it). The Einstein-Hilbert action then is written in terms of numbers of triangles and deficit angles, but there is no metric tensor, there is no gravitational field. In this language gravity is discussed in terms of geometry. If you put a field as an addition to the action, then the field is coupled to the geometry "minimally", as the field is defined on the underlying geometry, but there is no exact term like field times metrics. In this case there is no direct coupling constant measuring the strength of this interaction.
