Maxwell's demon (J.C. Maxwell): A contemplation experiment describing the concepts of entropy. We contain a container of gas that is partitioned into two equivalent sides; each side is in thermal equilibrium with the other. The walls and the separation of the container are ideal insulators.
Now suppose there is a very small demon who is waiting at the separation next to a small trap door. He can close and open the door with slight (negligible) work. Let's state he opens the door to permit a fast-moving molecule to travel from the left side to the right, or for a slow-moving molecule to travel from the right side to the left, and remains it closed for all other molecules. The total effect would be a flow of heat -- from left side to the right -- even although the container was in thermal equilibrium. This is obviously a violation of the second law of thermodynamics.
Thus where did we go wrong? It turns out that information has to do with entropy also. In order to sort out the molecules according to speeds, the demon would be containing to keep a memory of them -- and it turns out that rise in entropy of the maintenance of this simple memory would be more than make up for the reduction in entropy due to the flow of heat.