From BlenderWiki

Median Point as Pivot

We can assimilate the “median point” to the notion of Center Of Gravity (COG): supposing that every element of the selection has the same mass, the median point would sit at the COG, the point of equilibrium for the selection. This is very abusive as we will see soon enough. Yet it helps predicting where the median point should be when planing a scene.

In Object mode

For objects, only the object center is taken into account. Moreover, each object center is assumed to have the same mass. This can lead to very counter-intuitive results. On the (Median point of object centers and ObData), we see that the median of the objects sits far from the middle of the object centers. That is because the ObData (the geometry) of the moon and the star is way off their object center.

In Edit mode

Still on (Median point of object centers and ObData), we see that even the position of the median point for the ObData is surprisingly close to the hourglass: this is because it has much more vertices (611) than the moon (81) and the star (130). Blender supposes that every vertex has the same weight.

Transformation

Once the Median Point has been chosen from the list, the transform manipulator (if enabled) immediately cling to it, giving and excellent visual clue: all the rotations, scalings and mirror will happen around this point.