A stationary stochastic process is one that has a probability distribution that does not change with shifts in time (or space). Most of the math associated with stochastic processes requires the underlying process to be stationary for the math to work.
Unfortunately this is not necessarily true with socio-economic systems and there is not enough of a vocabulary to go with the notion of non-stationary processes. It is difficult to begin explaining the notion of stationarity every time I simply want to say that a recession in a modern environment need not resemble a recession in the last decade. The whole underlying process could have changed, along with the probability distribution. In an earlier post I talked about how systems built on top of one another will exacerbate the fat-tailed effects - that is nothing but a change in the distribution, a non-stationary process.
This applies to surprisingly homely notions which I will discuss later. A lot of problems can arise out of wrongly assuming a stationary process, even in down-to-earth topics.