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.
