In 1889 G.L. Gouy found that the "Brownian" movement was more rapid for smaller particles (we do not notice Brownian movement of cars, bricks, or people). In 1900 F.M. Exner undertook the first quantitative studies, measuring how the motion depended on temperature and particle size.
The first good explanation of Brownian movement was advanced by Desaulx in 1877: "In my way of thinking the phenomenon is a result of thermal molecular motion in the liquid environment (of the particles)." This is indeed the case. A suspended particle is constantly and randomly bombarded from all sides by molecules of the liquid. If the particle is very small, the hits it takes from one side will be stronger than the bumps from other side, causing it to jump. These small random jumps are what make up Brownian motion.
The first mathematical theory of Brownian motion was developed by Einstein in 1905.
References: Encyclopedia Brittanica 1968, "Brownian Movement."