How should I use the complex permittivity of a material? Here:
$$\epsilon = \epsilon' + j *\epsilon''$$
I understand that the first part ($\epsilon'$) is the relative permittivity of a material, while the second part
$\epsilon'' = \frac{\sigma}{\epsilon_0\omega}$
is taking into account the dielectric loss due to conductivity($\sigma$), while $\omega$ takes into account the frequency-dependence of $\epsilon'$ and $\sigma$.
What I do not understand is which value of complex permittivity do I use in an equation that involves $\epsilon$?

*

*Do I use the absolute magnitude?
In this case, the resultant will always be greater than $\epsilon'$, which I'm not sure makes sense.

*Do I use the complex permittivity as is, and ultimately take the real/complex parts as needed?
In this case, I'm not sure what each part physically means.

 A: 
Do I use the complex permittivity as is, and ultimately take the real/complex parts as needed?

You can do that. The fields are also complex, and the physical part is often obtained by taking the real part, but don't mistake the real part of a product for the product of real parts.
For example, if $D$ and $E$ are the complex electric displacement and electric field, respectively, you can use the complex $\epsilon$ in the equation:
$$
D = \epsilon E
$$
and
$$
Re(D) = Re(\epsilon)Re(E)-Im(\epsilon)Im(E)
$$

In this case, I'm not sure what each part physically means.

The real part is related to what we usually call the "index of refraction," often denoted by $n$.
The imaginary part is related to the absorption, often denoted by $\mu$.

You can also relate the complex $\epsilon$ to various other "optical constants" such as the polarizability $\alpha$ like:
$$
\epsilon = 1 + 4\pi\alpha\;,
$$
(in cgs Gaussian units).
As another cute example, you can also relate $\epsilon$ to the "plasma frequency" $\omega_p$ like:
$$
\frac{\pi}{2}\omega_p^2 = -\int_0^\infty d\omega \omega Im(1/\epsilon(\omega))\;.
$$
It's also fun to study properties of the complex dielectric function $\epsilon(\omega)$ as a function of complex $\omega$. For example, you can show that $\epsilon(\omega)$ has no poles in the complex upper half plane.
