Brownian motion is a fundamental stochastic process characterized by continuous, random, and independent increments with normally distributed changes over time, whose variance grows proportionally with the time interval. It serves as a mathematical model for diverse natural and financial phenomena, exhibiting key properties such as the Markov property, reflection principle, quadratic variation, and extensions like Brownian motion with drift, reflected and absorbed Brownian motions, and the Brownian bridge, all crucial for understanding random dynamics and derivative pricing.