A silent acknowledgement and understanding between two people, who are both wishing or thinking the same thing (and are both unwilling to initiate).
My guess is that most people have experienced this, and although I cannot begin to pronounce it, I think it is a pretty cool looking word. (Spellcheck has had it's way with me the last few days... Take that- Spellcheck!)
Examples I can come up with when mamihlapinatapai might happen...
Looking into the eyes of a loved one when you both know hospice is the next step.
A leader's expression with his or her staff when it's time to declare war.
Spouses that cannot get along realizing that divorce is inevitable.
Capturing that acknowledgement for me is most easily seen watching from the outside, when I myself don't have to be participant in mamihlapinatapai, but as a witness to it in the expressions of others.