A Mexican face-off (or otherwise known as a Mexican Standoff) takes its name from concept element in many modern action movies and takes the fundamental definition of a face-off (equal tactical advantage preventing action) to extreme, inter-personal close quarters. It is commonly used by movie directors to break an action scene so that dialogue can take place.
Mexican face-off's occur when two characters aggressive toward each other end up at close quarters, where both are able and ready to kill each other should one of the two parties initiate the next attack. Typically firearms appear in such situations with each character pointing their muzzle at a part of the other character's body where, if shot, the bullet wound would cause instant fatality. However, the same rules of the Mexican face-off can be applied with bladed weapons, or explosives in some cases.
When faced with a Mexican face-off, characters are forced not to deal the killing blow as the reprisals from their foe would cause their own death.
The movie Face-Off saw Sean Archer and Castor Troy (where both Nicolas Cage and John Travolta assumed both roles throughout the movie) enter into many Mexican face-off's, one including more than two people.
Although Saving Private Ryan is considered as more of a historical war drama than an action movie, it includes a Mexican face-off, where the Allied task-force threaten and are threatened by Axis troops when a wall collapses.
If you are ever involved in a Mexican face-off, you are advised to retreat to cover while maintaining a good line of sight to your agressor, and shoot only when you have the tactical advantage. Carelessnes costs lives.