Definition: Coupling What does it mean to say that 2 fields are coupled? More generally, what does "coupling" mean?
 A: "Coupling" is just a particle physicist's (but also a rather generic other physicist's) favorite word for an "interaction". 
In a theory given by a Lagrangian, degrees of freedom (e.g. fields but not necessarily fields) $f,g$ are coupled if there exist terms in the Lagrangian that depend both on $f$ and $g$, typically (in field theory) on their product or a product of their derivatives. This term in the Lagrangian – e.g. $e\cdot \bar\Psi\gamma^\mu \Psi\cdot A_\mu$ in QED – is called a "coupling" by itself (in my example, a coupling between electrons and photons). The coefficient of such a term (e.g. $e$ in my example) is often called the "coupling constant", at least in certain conventions.
In the opposite case, $f,g$ are said to be "decoupled": they don't interact with each other. The action may be divided to an action that only depends on the $f$-degrees of freedom and those that only depend on the $g$-degrees of freedom which means that the minimization proceeds separately and these two degrees of freedom evolve independently of one another.
The word "coupling" became widespread long before quantum field theory. In its "toy model", a multi-dimensional harmonic oscillator, one may write terms such as $K x_1 x_2$ in the Lagrangian which will make the two springs with coordinates $x_1$ and $x_2$ interact (coupled): the energy will be transmitted from one to the other and back.
