A strictly stationary process (or time series) is one whose joint distribution is constant over time. That is, the joint distribution of any set of $k+1$ observations $\{x_t, ..., x_{t+k}\}$ does not depend on $t$. So the process "looks the same" probabilistically wherever you are in time.
A weakly stationary process or series is one whose mean, $E(x_t)$, and covariance function, $\text{Cov}(x_t, x_{t+k})$ (variance and autocorrelation function), are constant over time.