the consistency condition
Suppose you want to build a process from the ground up by writing down its finite-dimensional distributions — the joint law of the values at every finite list of times. You have a lot of freedom, but not total freedom: the snapshots you specify must agree with one another wherever they overlap. The consistency conditions are exactly the two common-sense rules that say 'these snapshots fit together into one coherent process'.
There are two rules. First, permutation consistency: the joint law of (X_1, X_2) and the joint law of (X_2, X_1) must be the same picture with the coordinates relabelled — reordering the time labels cannot change the underlying probabilities. Second, marginalization consistency: if you describe (X_1, X_2, X_3) and then forget about X_3 by summing or integrating it out, what you get must match the law you separately specified for (X_1, X_2). In words: a smaller snapshot must be recoverable as a margin of a larger one. A family of finite-dimensional distributions obeying both is called consistent.
Why does this matter? Because of a remarkable payoff (Kolmogorov's extension theorem): if a family of finite-dimensional distributions is consistent, then there actually exists a genuine stochastic process — defined on a single probability space, with sample paths and all — whose finite snapshots are exactly the ones you wrote. Consistency is the precise hypothesis that licenses you to assemble an infinite-dimensional object from finite pieces. Without it the pieces would contradict each other and no such process could exist.
Suppose you declare X_1 and X_2 are each standard normal, that (X_1, X_2) is jointly normal with correlation 0.5, and also that (X_1, X_2, X_3) is jointly normal with a given covariance matrix. Consistency demands that integrating X_3 out of the three-variable law reproduces exactly the two-variable law you already gave for (X_1, X_2). If those two specifications clash, your family is inconsistent and no process can realize it.
Smaller snapshots must be margins of larger ones, and time labels may be reordered freely — those are the two consistency rules.
Consistency guarantees a process exists, but it does NOT guarantee a continuous-path version — Kolmogorov's extension gives existence on a function space, while continuity of paths needs an extra, separate theorem (the Kolmogorov continuity criterion).