Just to refresh anyone who may have forgotten
Intuitive metaphor for Brownian motion
Consider a large balloon of 10 meters in diameter. Imagine this large balloon in a football stadium or any widely crowded area. The balloon is so large that it lies on top of many members of the crowd. Because they are excited, these fans hit the balloon at different times and in different directions with the motions being completely random. In the end, the balloon is pushed in random directions, so it should not move on average. Consider now the force exerted at a certain time. We might have 20 supporters pushing right, and 21 other supporters pushing left, where each supporter is exerting equivalent amounts of force. In this case, the forces exerted from the left side and the right side are imbalanced in favor of the left side; the balloon will move slightly to the left. This imbalance exists at all times, and it causes random motion. If we look at this situation from above, so that we cannot see the supporters, we see the large balloon as a small object animated by erratic movement.
That's the Wikipedia definition and very apt too imho. Now add some crowd psychology and I think we may be getting somewhere. The people below the balloon would probably want the damn thing to move off them in any direction. After all how much fun is it being suffocated by a giant balloon. So if the balloon moves to the left a little then the people on both sides, everything else being equal, would also push it to the left and it would go left at an increasing speed.
From balloon read stock/index price and one has a clearer picture of what is happening. Many a time the price is in a sort of limbo before being propelled up or down.