Consider an ODE method applied to the model problem
$\qquad \dfrac{dy}{dt} = \lambda y \quad y(0)=y_0 \qquad (1)$
the region of absolute stability of the ODE method is the region in the complex plane, $\mathcal{R}$, such that if $dt\lambda \in \mathcal{R}$ then $y(t) \to 0 $ as $t \to \infty$ for all initial values $y_0$. The region of absolute stability is a property of the ODE method. When solving ODE's, knowing the region of absolute stability for the ODE method is useful for estimating the timestep size required to obtain qualitatively correct solutions.
Consider the standard Runge-Kutta methods of orders 1 through 4. When applied to the model problem (1), the resulting difference equations are
$\begin{array}{c|r|l}RK1 & y_{k+1} &= (1 + dt\lambda) y_k \\ RK2 & y_{k+1} &= (1 + dt\lambda + \dfrac{{(dt\lambda)}^2}{2}) y_k \\ RK3 & y_{k+1} &= (1 + dt\lambda + \dfrac{{(dt\lambda)}^2}{2} + \dfrac{{(dt\lambda)}^3}{6} ) y_k \\ RK4 & y_{k+1} &= (1 + dt\lambda + \dfrac{{(dt\lambda)}^2}{2} + \dfrac{{(dt\lambda)}^3}{6} + \dfrac{{(dt\lambda)}^4}{24} ) y_k \end{array}$
Thus, the regions of absolute stability for the standard RK methods are defined by the regions in the complex plane such that
$\begin{array}{c|l}{\rm RK1} & |(1 + z)| < 1 \\ RK2 &| (1 + z + \dfrac{{z}^2}{2})| < 1 \\ RK3 & | (1 + z + \dfrac{{z}^2}{2} + \dfrac{{z}^3}{6} )| < 1\\ RK4 & | (1 + z + \dfrac{{z}^2}{2} + \dfrac{{z}^3}{6} + \dfrac{{z}^4}{24} ) |< 1 \end{array}$
See "Stability Regions For Runge-Kutta Method of Order One to Four" for plots of these regions.