a sample path (realization)
Run the random experiment once — let the whole universe of chance settle into a single outcome — and watch what the process actually did over time. The curve you record is a sample path: one complete trajectory, the process as it happened on this particular run. It is the thing you would actually plot if you ran a simulation once.
Formally, fix a single outcome omega of the underlying randomness. Then for that fixed omega the family { X_t } stops being random and becomes an ordinary, definite function of t alone — t maps to X_t(omega). That function is the sample path for omega. Every different omega gives a different path; the process is the entire ensemble of paths, each carrying the probability of its omega. So a sample path is to a process what a single number is to a random variable: one realized value, drawn at random — except here the 'value' is a whole function.
This is where path-level questions live, the ones a single marginal can never answer: Is the path continuous, or does it jump? Does it ever cross zero? What is the maximum it reaches before time T? How long does it spend positive? Two processes can have identical one-time-snapshot distributions yet wildly different paths — one smooth, one wildly jagged — so the law of the paths, not just the marginals, is what truly pins a process down.
Simulate a coin-flip random walk once: you might get the path 0, 1, 0, -1, 0, 1, 2, ... — one specific zigzag staircase. Simulate again and you get a different zigzag. Each individual zigzag is a sample path; the full random walk is the collection of all possible zigzags, weighted by their probabilities.
A sample path is one whole trajectory from a single run; the process is the ensemble of all of them.
Beware the common slip of identifying 'the process' with 'a sample path'. A single observed trajectory is one draw, not the process itself — just as one rolled 4 is not the die. Path-level properties (continuity, smoothness) are properties of the law, not of any one path you happened to see.