GARCH specification - why are $\sigma_t^2$ and $\epsilon_t^2$ not the same? Often times people specify the GARCH model as follows:
$$ \sigma _{t}^{2}=\omega +\alpha _{1}\epsilon _{t-1}^{2}+\cdots +\alpha _{q}\epsilon _{t-q}^{2}+\beta _{1}\sigma _{t-1}^{2}+\cdots +\beta _{p}\sigma _{t-p}^{2}=\omega +\sum _{i=1}^{q}\alpha _{i}\epsilon _{t-i}^{2}+\sum _{i=1}^{p}\beta _{i}\sigma _{t-i}^{2}.$$
I must admit that I am a little confused as to what the difference is between $\sigma_t^2$ and $\epsilon_t^2$. As far as I (though I did) know the squared error terms equal the variance, because of the fact that its mean equal zero. 
In other words; what I am wondering is what is the actual "numbers" that is put into $\sigma_t^2$'s place in the equation when estimating?
 A: In a GARCH model for a time series $x_t$ we have 
\begin{aligned}
x_t&\sim i.i.D(\mu_t,\sigma_t^2), \\
\mu_t&=... \text{ (conditional mean of } x_t \text{ given past information)} \\
\sigma_t^2&=\omega +\sum _{i=1}^{q}\alpha _{i}\epsilon _{t-i}^{2}+\sum _{i=1}^{p}\beta _{i}\sigma _{t-i}^{2} \text{ (conditional variance of } x_t \text{ given past information)} 
\end{aligned}
where $D$ is some distribution parameterized by the conditional mean $\mu_t$ and conditional variance $\sigma_t^2$, and $\epsilon_t:=x_t-\mu_t$ is an additive error term, a random variable itself. Meanwhile, $\sigma_t^2$ is the conditional variance of $x_t$ and simultaneously of $\epsilon_t$, hence, a parameter (an unknown constant). You are right that $\mathbb{E}(\epsilon_t^2)=\sigma_t^2$, but that does not make for $\epsilon_t^2=\sigma_t^2$.
When we are estimating the model, we treat $\sigma_t^2$ as an unknown parameter and estimate it along with the other parameters such as $\omega$, $\alpha$s and $\beta$s. We are not putting in any numbers in for $\sigma_t^2$ in estimation because conditional variances are unobservable and are never a variable in our dataset (just like the other model parameters).
