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. For more information see stationary process.
Several sub-types of stationarity are defined: first-order, second-order, nth-order, wide-sense and strict-sense. For details please see the reference below.
An ergodic process is one which conforms to the ergodic theorem. The theorem allows the time average of a conforming process to equal the ensemble average. 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. Please see ergodic theory.