I'm just going to apologize first thing, because I know my understanding of these topics is very lacking.
I'm reading some lecture notes from what appears to be an econometrics course, and they are going over the stationarity of processes. In the course of defining stationarity, they provided the following definition of the autocovariance function:
$$ \gamma(s,t) = Cov(X_s,X_t)$$
They went on to say that for a stationary process, we have the following:
$$ \gamma_X(s,t) = \gamma_X(s+h,t+h) \forall s,t,h,\in \mathbb{Z} $$
and that because of this property, we can rewrite the autocovariance function as
$$ \gamma_X(h) = Cov(X_t, X_t+h) \text{ for } t,h\in\mathbb{Z}$$
I am only familiar with the latter definition of autocovariance. I am confused as to what could be meant by the former, in the case that $\{X_t\}$ is a non-stationary process. Because we're dealing with time series, does it make sense to say "the covariance of $X_t$ and $X_s$?" There will only be one realization of $X$ at time $t$ or $s$, and furthermore only one realization of $X$ that necessarily has the same distribution as $X_t$, so how can we speak of the covariance of $X_s$ and $X_t$?
I'm sorry if this is worded in a confusing way.