The Ham Sandwich Cut is a line that divides a set of points so that half of the points are above the line and half of the points are below it.  This also works in higher dimensions and for other sets of objects other than points.  Suppose we had 2 sets of points in 3D, then we would have to use a plane to divide the points in half.

Why do we call it the Ham Sandwich Cut?
Suppose you and your brother (sister, or whoever) were given a ham sandwich, and you each wanted half.  Now you know that you won't be happy unless you each get the same amount.  But by the ham sandwich cut theorem there is always a way to cut the sandwich so that each person gets exactly half of the ham and half of the bread.  (even if the ham is crooked on the bread!)