ARMA
Consider $y_t$ that follows an ARMA($p,q$) process. Suppose for simplicity it has zero mean and constant variance. Conditionally on information $I_{t-1}$, $y_t$ can be partitioned into a known (predetermined) part $\mu_t$ (which is the conditional mean of $y_t$ given $I_{t-1}$) and a random part $u_t$:
\begin{aligned}
y_t &= \mu_t + u_t; \\
\mu_t &= \varphi_1 y_{t-1} + \dotsc + \varphi_p y_{t-p} + \theta_1 u_{t-1} + \dotsc + \theta_q u_{t-q} \ \ \text{(known, predetermined)}; \\
u_t | I_{t-1} &~\sim D(0,\sigma^2) \ \ \text{(random)} \\
\end{aligned}
where $D$ is some density.
The conditional mean $\mu_t$ itself follows a process similar to ARMA($p,q$) but without the random contemporaneous error term:
$$
\mu_t = \varphi_1 \mu_{t-1} + \dotsc + \varphi_p \mu_{t-p} + (\varphi_1 + \theta_1) u_{t-1} + \dotsc + (\varphi_m + \theta_m) u_{t-m}, $$
where $m:=\max(p,q)$; $\varphi_i=0$ for $i>p$; and $\theta_j=0$ for $j>q$. Note that this process has order ($p,m$) rather than ($p,q$) as does $y_t$.
We can also write the conditional distribution of $y_t$ in terms of its past conditional means (rather than past realized values) and model parameters as
\begin{aligned}
y_t &\sim D(\mu_t,\sigma_t^2); \\
\mu_t &= \varphi_1 \mu_{t-1} + \dotsc + \varphi_p \mu_{t-p} + (\varphi_1 + \theta_1) u_{t-1} + \dotsc + (\varphi_m + \theta_m) u_{t-m}; \\
\sigma_t^2 &= \sigma^2,
\end{aligned}
The latter representation makes the comparison of ARMA to GARCH and ARMA-GARCH easier.
GARCH
Consider $y_t$ that follows a GARCH($s,r$) process. Suppose for simplicity it has constant mean. Then
\begin{aligned}
y_t &\sim D(\mu_t,\sigma_t^2); \\
\mu_t &= \mu; \\
\sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dotsc + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2; \\
\frac{u_t}{\sigma_t} &\sim i.i.D(0,1), \\
\end{aligned}
where $u_t:=y_t-\mu_t$ and $D$ is some density.
The conditional variance $\sigma_t^2$ follows a process similar to ARMA($s,r$) but without the random contemporaneous error term.
ARMA-GARCH
Consider $y_t$ that has unconditional mean zero and follows an ARMA($p,q$)-GARCH($s,r$) process. Then
\begin{aligned}
y_t &\sim D(\mu_t,\sigma_t^2); \\
\mu_t &= \varphi_1 \mu_{t-1} + \dotsc + \varphi_p \mu_{t-p} + (\varphi_1 + \theta_1) u_{t-1} + \dotsc + (\varphi_m + \theta_m) u_{t-m}; \\
\sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dotsc + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2; \\
\frac{u_t}{\sigma_t} &\sim i.i.D(0,1), \\
\end{aligned}
where $u_t:=y_t-\mu_t$; $D$ is some density, e.g. Normal; $\varphi_i=0$ for $i>p$; and $\theta_j=0$ for $j>q$.
The conditional mean process due to ARMA has essentially the same shape as the conditional variance process due to GARCH, just the lag orders may differ (allowing for a nonzero unconditional mean of $y_t$ should not change this result significantly). Importantly, neither has random error terms once conditioned on $I_{t-1}$, thus both are predetermined.
