I know that ARMA specifies the conditional mean of the stochastic process and that GARCH specifies the conditional variance of the stochastic process. As you can see in the answer to this question here, https://stats.stackexchange.com/questions/41509/what-is-the-difference-between-garch-and-arma.
However, I was wondering what this means on a more heuristic level as to when would be most appropriate to apply the GARCH model or the ARMA/ARIMA model to a particular times series? What features would the times series exhibit say or whether this is case of what you would like to forecast about the time series.
As you see in the linked answer, there is no contradiction between ARMA models and GARCH models. You can fit the same time series with ARMA$(p,q)$ for the mean and GARCH$(p', q')$ for the variance of the error term. Basically, if you have a heteroscedastic variance, then it may me helpful to model these dynamics. For instance, when you are interested in the "volume of trade" or in the "risk" of some asset, then you should model the variance of the time series. While, for the ARMA terms you are mainly interested in the values of the series itself, rather than in its variance. Namely, you want to forecast the value of some asset at time $t+1$, hence you should model the mean response of the process $\{Y_t\}_{t\ge 0}$ .