I was wrestling with stationarity in my head for a while... Is this how you think about it? Any comments or further thoughts will be appreciated.
Stationary process is the one which generates time-series values such that distribution mean and variance is kept constant. Strictly speaking, this is known as weak form of stationarity or covariance/mean stationarity.
Weak form of stationarity is when the time-series has constant mean and variance throughout the time.
Let's put it simple, practitioners say that the stationary time-series is the one with no trend - fluctuates around the constant mean and has constant variance.
Covariance between different lags is constant, it doesn't depend on absolute location in time-series. For example, the covariance between t and t-1 (first order lag) should always be the same (for the period from 1960-1970 same as for the period from 1965-1975 or any other period).
In non-stationary processes there is no long-run mean to which the series reverts; so we say that non-stationary time series do not mean revert. In that case, the variance depends on absolute position in time-series and variance goes to infinity as time goes on. Technically speaking, auto-correlations to not decay with time, but in small samples they do disappear - although slowly.
In stationary processes, shocks are temporary and dissipate (lose energy) over time. After a while, they do not contribute to the new time-series values. For example, something which happened log time ago (long enough) such as World War II, had an impact, but, it the time-series today is the same as if World War II never happened, we would say that shock lost its energy or dissipated. Stationarity is especially important as many classical econometric theories are derived under the assumptions of stationarity.
A strong form of stationarity is when the distribution of a time-series is exactly the same trough time. In other words, the distribution of original time-series is exactly same as lagged time-series (by any number of lags) or even sub-segments of the time-series. For example, strong form also suggests that the distribution should be the same even for a sub-segments 1950-1960, 1960-1970 or even overlapping periods such as 1950-1960 and 1950-1980. This form of stationarity is called strong because it doesn't assume any distribution. It only says the probability distribution should be the same. In the case of weak stationarity, we defined distribution by its mean and variance. We could do this simplification because implicitly we assumed normal distribution, and normal distribution is fully defined by its mean and variance or standard deviation. This is nothing but saying that probability measure of the sequence (within time-series) is the same as that for lagged/shifted sequence of values within same time-series.