Aaaaah! They said WHAT?!
Here is a good place to start on http://en.wikipedia.org/wiki/Brownian_motion" .
On the scale of individual molecules, particles are rapidly undergoing collisions for all of the other particles. Its like bumper cars. A particle will get bumped in one direction, then it will get bumped in another, and another... and you end up with 'brownian motion' or the 'random walk'.
On larger scales, like small dust particles, brownian motion can still be important. For particles as large as dust, TONS and TONS of particles are ALWAYS bumping into them from all sides. Every once and a while they get bumped a little extra hard in one direction, however, and then a fraction of a second later, they get bumped extra hard in another direction----thus they also can undergo brownian motion.
Hope that helps; and let me know if you still have questions.