Let me expand a little more on what Craig Thone just said :

Consider the energy/frequency-dependent Green function :
$$
\tilde{G}(\omega)=\frac{1}{\omega-(a-\mathrm{i}b)}
$$
with one single pole in $\omega=a-\mathrm{i}b$ (with $b>0$), which is Fourier transform of the time-dependent $G(t)$ Green function such as :
$$
G(t)=\int\frac{\mathrm{d}\omega}{2\pi}\frac{e^{-\mathrm{i}\omega t}}{\omega-(a-\mathrm{i}b)}
$$
One can show, using complex analysis (I can eventually show some details about that if needed), that it computes to :
$$
G(t)=\mathrm{i}\,e^{\mathrm{i}at-bt}\Theta(t)
$$
where $\Theta$ is an Heaviside step-function.

It means that :
> 1. The real part $a$ of the pole gives an oscillatory behavior to the solution. In the context of quantum systems, $a$ is often referred to as an eigen-energy.
2. The role the imaginary part $b$ in twofold :
     - It gives a damped behavior to the solution. In the context of quantum systems, $b$ will describe how one state, *which is not an eigen-state of the system*, will be depleted as a function of time in a superposition of eigen-states. In the limit of a [perturbation theory](https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)), $b$ is the same rate given by the [Fermi Golden Rule](https://en.wikipedia.org/wiki/Fermi%27s_golden_rule).
     - The fact that $b>0$ ensures that the function $\tilde{G}(\omega)$ has no pole in the upper complex plane (i.e. $\forall\omega\in\mathbb{C}, \Im({\omega})>0$). This analycity of $\tilde{G}$ implies by the [Cauchy's integral theorem](https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem) that :
$$
\forall t<0,\,G(t)=0
$$
which is needed to ensure the *causality* of the dynamics. To ensure such property, an Heaviside step-function $\Theta$ can be added to $G(t)$.

**EDIT** : **Analyticity of $G$ and causality**.
In order to evaluate $G(t)$ for $t<0$, one can consider the path $\Gamma_R$ which is defined as :
$$
\Gamma_R=\left\{\omega\in\mathbb{C},\,\omega\in[-R,R]\,\bigcup\,\mathcal{C}_R^0\right\}
$$
where $\mathcal{C}_R^0$ is half of the circle centered on $\omega=0$ with a $R$ radius.

$\forall t<0,\, G(t)$ is given by the integral :
$$
\forall t<0,\, G(t)=\lim_{R\rightarrow+\infty}\int_R^{-R}\frac{\mathrm{d}\omega}{2\pi}\frac{e^{-\mathrm{i}\omega t}}{\omega-(a-\mathrm{i}b)}
$$
Additivity of integrals gives you then :
$$
\forall t<0,\,\int_{\Gamma_R}\frac{\mathrm{d}\omega}{2\pi}\frac{e^{-\mathrm{i}\omega t}}{\omega-(a-\mathrm{i}b)}=G(t)+\lim_{R\rightarrow+\infty}\int_{\mathcal{C}_R^0}\frac{\mathrm{d}\omega}{2\pi}\frac{e^{-\mathrm{i}\omega t}}{\omega-(a-\mathrm{i}b)}
$$
Since there is no pole in the complex area enclosed by $\Gamma_R$ ($\tilde{G}$ is analytic in the upper complex plan), Cauchy's integral theorem gives you that :
$$
\int_{\Gamma_R}\frac{\mathrm{d}\omega}{2\pi}\frac{e^{-\mathrm{i}\omega t}}{\omega-(a-\mathrm{i}b)}=0
$$
Moreover, [Jordan's lemma](https://en.wikipedia.org/wiki/Jordan%27s_lemma) ensures that :
$$
\lim_{R\rightarrow+\infty}\int_{\mathcal{C}_R^0}\frac{\mathrm{d}\omega}{2\pi}\frac{e^{-\mathrm{i}\omega t}}{\omega-(a-\mathrm{i}b)}=0
$$
letting you with :
$$
\forall t<0,\,G(t)=0
$$
You have to realize this is the reason why sometimes you see people introduce some infinitesimal $\pm\mathrm{i}\epsilon$ which pushes the pole of a [propagator](https://en.wikipedia.org/wiki/Propagator#Relativistic_propagators) out of the real axis : it ensures causality/anti-causality of the solution.