What does it mean to build a so-called mean equation? A series is denoted by Xt. How do we build a time-series model for Xt that is the "mean equation" of Xt? Is this another way of asking to build a stationary time-series?
 A: A reference to the mean model refers to a model for the conditional expectation. Consider specifying a model for the stochastic process $(Y_t)$. You can specify a model for any of the unconditional or conditional features of the stochastic process. 
The most common kinds of stochastic models are models of the conditional expectation or conditional mean of the stochastic process, $\mathbb{E}(Y_t \mid \mathcal{I}_t)$, where the conditioning is in terms of other information that is known at time $t$.  
The notion of information known at time $t$ is captured by the information set, $\mathcal{I}_t$ (which is the $\sigma$-algebra induced by past observations, but that point can be glossed over for now). 
Simple models for the conditional mean are linear in elements of the conditioning set
$$
\mathbb{E}(Y_t \mid \mathcal{I}_t) = \alpha_1 Y_{t-1} 
$$
is the AR(1) model for example.
You can imagine models for other features of the distribution conditional on information known at time $t$, such as a model for the conditional variance $\mathbb{V}(Y_t \mid \mathcal{I}_t)$ (GARCH models), or even more exotic distributional features such as the conditional kurtotis or conditional skewness.
