Brownian motion is the random movement of particles suspended in a fluid. It is caused by random bombardment by molecules of the fluid.
Mathematically, a Brownian motion process is such that if we know a particle's position is p at time 0, then the particle's position at subsequent time t is a normally distributed random variable with a mean of p and a variance of t. Brownian motion is a kind of stochastic process.
The mathematical theory of Brownian motion has been applied in contexts ranging far beyond the movement of particles in fluids. For example, in the modern theory of option pricing, asset classes are sometimes modelled as if they move according to a Brownian motion with drift.