The number of eigenvalues is equal to the number of state variables. In our case there will be 2 eigenvalues. Eigenvalues are generally complex numbers. If real parts of all eigenvalues are negative, then the equilibrium is stable. If at least one eigenvalue has a positive real part, then the equilibrium is unstable.
Eigenvalues have the same meaning as the slope of a line in phase plots.
Negative real parts of eigenvalues indicate a negative feedback. It is important that ALL eigenvalues have negative real parts. If one eigenvalue has a positive real part then there is a direction in a 2-dimensional space in which the system will not tend to return back to the equilibrium point. Overcompensation means that the system jumps over the equilibrium point because the negative feedback is too strong. Then it returnes back and again jumps over the equilibrium.