Brownian motion is, according to Wikipedia, the "random motion of particles suspended in a fluid (be that a liquid or a gas) resulting from their collision with the fast-moving atoms or molecules in the gas or liquid."
This was named after Robert Brown (who would have guessed "Brownian theory" was named after someone named "Brown"?). Anyway, in 1827, while squire Brown was peering through a microscope at particles trapped in cavities inside pollen grains in water, he made note that the particles moved through the water. He saw no mechanism for this movement. Why did they move? No answers at first.
However, in 1905, physicist Albert Einstein, in an epic move to be a smart aleck, published a paper that explained how motion that Brown observed occurred. It occurred by the pollen being moved by the water molecules. Later on, someone even smugger than Einstein actually received a Nobel Prize, in 1908, for related work on the subject: French physicist Jean Perrin, He noted that the direction of the force of the atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. Can you believe they gave the guy a Nobel Prize for that? Well, it was pretty clever.
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