Brownian motion is a phenomena whereby small particles suspended in a liquid tend to move in pseudo-random or stochastic paths through the liquid, even if the liquid in question is calm. It is the result of asymmetry in the kinetic impacts of molecules that make up the liquid. The liquid phase, by definition, must have some temperature, meaning its molecules or atoms must be thermally excited, bumping into each other and objects suspended within them. To picture this phenomena, a person can imagine the motion of golf balls on a table filled with thousands of ball bearings moving in quick trajectories.
The phrase Brownian motion can also refer to mathematical models used to describe the phenomenon, which have considerable detail and are used as approximations of other stochastic motion patterns. The mathematical motion is related to, but more structured than, the random walk, in which the displacement of a particle is entire randomized. The phenomena has the Markov property, a term from probability theory which means that the future state of the particle is determined entirely by its current state, not by any past state. Used in this sense, the mathematical concept is slightly different, but very similar to, physical Brownian motion.
The scientist who made Brownian motion famous is Albert Einstein, who brought the phenomenon to the attention of the larger physics community by publishing a paper on it in 1905, his personal annus mirabilis or "wonderful year." The phenomenon was observed as early as 1765, but not described or studied in detail until the botanist Robert Brown's research in 1827,and it is named in honor of his work. As a botanist, Brown first observed the effect in pollen floating in water, where it is visible with the naked eye. Through experimentation, Brown determined that the specks of pollen were not propelling themselves independently, but rather that their motion was pseudo-random.
Jean Perrin, a French physicist who later won the Nobel prize, springboarded off of Einstein's work. Using Brownian motion as evidence, in 1911 he proved once and for all that matter is made of atoms and molecules. Although atomic theory is originally credited to John Dalton, the 18th- and 19th-century British physicist, it was under dispute for over a century, and it was Perrin's work that resulted in its universal acceptance.