terminology - What is a mode?

The word mode pops up in many fields of physics, yet I can't remember ever encountering a simple but precise definition.

After having searched fruitlessly on this site as well, I feel that even though it seems like a trivial question, an easy to find place with (one or more) good answers is in order.

Objective:

Ideally, answers should give a graspable, intuitive and easy-to-remember definition of what a mode is, preferably in a general context. If limitation is necessary for a more detailed answer, assume a context of theoretical physics, e.g. mode expansions in quantum field theory.

Answer

In a very mathematical sense, more often than not a mode refers to an eigenvector of a linear equation. Consider the coupled springs problem $$\frac{d}{dt^2} \left[ \begin{array}{cc} x_1 \\ x_2 \end{array} \right] =\left[ \begin{array}{cc} - 2 \omega_0^2 & \omega_0^2 \\ \omega_0^2 & - \omega_0^2 \end{array} \right] \left[ \begin{array}{cc} x_1 \\ x_2 \end{array} \right]$$ or in basis independent form $$ \frac{d}{dt^2}\lvert x(t) \rangle = T \rvert x(t) \rangle \, .$$ This problem is hard because the equations of motion for $x_1$ and $x_2$ are coupled. The normal modes are (up to scale factor) $$\left[ \begin{array}{cc} 1 \\ 1 \end{array} \right] \quad \text{and} \quad \left[ \begin{array}{cc} 1 \\ -1 \end{array} \right] \, .$$ These vectors are eigenvectors of $T$. Being eigenvectors, if we expand $\lvert x(t) \rangle$ and $T$ in terms of these vectors, the equations of motion uncouple. In other words

The set of normal modes is the vector basis which diagonalizes the equations of motion (i.e. diagonalizes $T$).

That definition will get you pretty far.

The situation is the same in quantum mechanics. The normal modes of a system come from Schrodinger's equation $$i \hbar \frac{d}{dt}\lvert \Psi(t) \rangle = \hat{H} \lvert \Psi \rangle \, .$$ An eigenvector of $\hat{H}$ is a normal mode of the system, also called a stationary state or eigenstate. These normal modes have another important property: under time evolution they maintain their shape, picking up only complex prefactors $\exp[-i E t / \hbar]$ where $E$ is the mode's eigenvalue under the $\hat{H}$ operator (i.e. the mode's energy). This was actually also the case in the classical system too. If the coupled springs system is initiated in an eigenstate of $T$ (i.e. in normal mode), then it remains in a scaled version of that normal mode forever. In the springs case, the scale factor is $\cos(\sqrt{\lambda} t)$ where $\lambda$ is the eigenvalue of the mode under the $T$ operator.

From the above discussion we can form a very physical definition of "mode":

A mode is a trajectory of a physical system which does not change shape as the system evolves. In other words, when a system is moving in a single mode, the positions of its parts all move with same general time dependence (e.g. sinusoidal motion with a single frequency) but may have different relative amplitudes.