A random process is a sequence of random variables, indexed on some set, which is typically the integers or real numbers for our purposes.
We will discuss the property of being stationary for random processes that are indexed by time (discrete-time or continuous-time). Stationarity refers to the time-invariance of a probability distribution. The definition can be extended to higher dimensions, where instead of time we are dealing with space, for purposes such as image processing. In those cases, stationarity refers to shift-invariance. A random process is stationary if the probability distribution of any finite subset of points is time-invariant. That is, shifting all time indices by the same amount results in a new set of random variables with the same distribution. Even if a process is not stationary in the strict sense, it may exibit some qualities of stationarity that matter for estimation. A process that is stationary in the second first and second moments is refered to as wide-sense stationary. Jointly stationary (or wide-sense stationary) processes are a collection of random processes that satisfy the same property as stationary (or WSS) processes, even when considering also joint distributions of variables from more than one sequence in the collection.
A sinusiod with a uniform random phase shift is stationary.
A uniform distribution over the four positive and negative sin and cosine functions is wide-sense stationary, although not stationary.
The autocorrelation function of a random process defines the random process captures the second moments of the random process. In our usage, it will refer simply to the correlation of pairs of variables. However, other define it to be the covariance or correlation coefficient. It won't really make a difference, especially since we will almost always assume the rnadom process has zero-mean. Read about these topics in the appendix (A 1.2.1) of Kay Vol. 1