For example, I stated this fact/theorem at a dinner to some friends and they were pretty impressed.
Given any sequence of n integers, positive or negative, not necessarily all different, some consecutive subsequence has the property that the sum of the members of the subsequence is a multiple of n.
wording taken from http://www.cut-the-knot.org/pigeonhole/group.shtml
I stated the above in laymans terms so they could understand it and they were interested in it (at least for a few minutes).
What are some other accessible theorems/facts similar to this that non-mathematicians can understand? or maybe simple applications of mathematics that they will understand?