independent and stationary increments
An increment is simply the change of a process over a time interval: how much X moved from time s to time t, written X_t - X_s. Many of the cleanest, most important processes are defined not by their values but by the behaviour of their increments — and two properties of increments are so powerful they almost single-handedly generate a process.
Independent increments means that changes over non-overlapping time intervals are statistically independent of one another: what the process does between times 1 and 3 tells you nothing about what it does between times 5 and 8. The future change does not depend on the past change. Stationary increments means the distribution of an increment depends only on the length of the interval, not on where it starts: X_{t+h} - X_s+... — more precisely, X_{t} - X_{s} has the same distribution as X_{t-s} - X_0, so an increment over any 2-second window has the same law no matter when the window sits. (Stationary increments is a weaker, increment-level cousin of full stationarity; a process with stationary increments need not itself be stationary.)
Together these two are the structural skeleton of the simplest processes. The random walk is the discrete prototype: its steps are independent and identically distributed, which is exactly independent and stationary increments for integer times. Let the steps shrink correctly and you get Brownian motion, the continuous prototype, whose increments are independent and stationary and Gaussian. The Poisson process is the counting prototype: independent, stationary increments with a Poisson distribution. Remarkably, independent and stationary increments plus continuity essentially force Brownian motion or a Levy process — so much structure follows from so little.
In a symmetric random walk, each step is +1 or -1 with probability 1/2, independently. The increment over steps 1 to 4 is the sum of three independent steps; the increment over steps 7 to 10 is the sum of three more, independent of the first and with the same distribution. Independent increments (different windows don't talk) and stationary increments (same-length windows have the same law) are both visible here.
Independent increments = non-overlapping changes are independent; stationary increments = a change's law depends only on the interval's length.
A process with stationary increments need NOT be a stationary process — Brownian motion has stationary, independent increments but its variance grows with time, so it is far from stationary. The two 'stationary' notions are different: one is about increments, the other about the values themselves.