Imagine a drunkard leaving a lamp post, each step equally likely to go left or right. Where will they be after a thousand steps? Not where they started — the distance from the lamp post grows, just not in a predictable direction. This is a random walk, and it describes stock prices, the diffusion of molecules, the path of a pollen grain in water.
The surprising thing is the square-root law: after n steps of size 1, the expected distance from the start is √n, not n. Doubling your time quadruples your uncertainty, not doubles it. A stock forecast for one year is not twice as reliable as one for four years — it is half as reliable.
Random walks also explain why past prices carry almost no information about future prices in efficient markets. Each step erases the memory of the last. The path looks meaningful in hindsight. It was not.