In essence this implies that the random process will not change its statistical properties with time and that its statistical properties (such as the theoretical mean and variance of the process) can be deduced from a single, sufficiently long sample (realization) of the process.
Stationarity is the property of a random process which guarantees that its statistical properties, such as the mean value, its moments and variance, will not change over time.
A stationary process is one whose probability distribution is the same at all times.
In practice this means that statistical sampling can be performed at one instant across a group of identical processes or sampled over time on a single process with no change in the measured result.
Although the measured process may be stationary in the long term, it is not appropriate to consider the sampled distribution to be the reflection of a single (ergodic) process: The ensemble average is meaningless.