A strictly stationary process (or time series) is one whose joint distribution is constant over time shifts. A weakly stationary (or covariance stationary) process or series is one whose mean and covariance function (variance and autocorrelation function) do not change over time.
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.
Strict stationarity does not imply weak stationarity (because mean, variance and/or autocorrelation of a strictly stationary process need not exist). Weak stationarity does not imply strict stationarity (because higher order moments of a weakly stationary process might be nonconstant over time).
Stationarity is an important concept in time series analysis. Time series data are often transformed to become stationary.
References:
Wikipedia - Stationary process
Investopedia - Stationary and nonstationary processes
