A "gringo" is a derogatory term which some Mexicans use to refer to Americans of Anglo-Saxon ethnicity. A "beaner" is a derogatory term which some Americans of Anglo-Saxon ethnicity use to refer to Mexicans.

The previous replies answered the question correctly. As an American living in Mexico, I can add this personal perspective: I have almost never heard the term "beaner" used by anyone in the U.S. However, the word "gringo" is use very frequently in Mexico. I also believe that "beaner" is a more offensive term than "gringo", as a matter of fact I do not find it offensive at all when people refer to me as a "gringo".

The word "gringo" comes from the fight between Mexicans and Americans. During the war 1845-1847 Mexico-American U.S. soldiers who invaded Mexico, sang a song called "green grow the lilacs" in English "Green Grow the Lilacs" and there comes the word gringo as a deformation of that sentence. Contemptuously the Americans say "beaner" to the Mexicans and it means "frijolero" "beaner", a term used in America to refer in a contemptuous colloquial to the Mexicans, while "gringo" is used by their counterparts to talk about Americans.

Yes))) I heard them in this song=)))

Do you know the song 'Frijolero' by Molotov? It has this gringo-beaner reference :)) http://www.youtube.com/watch?v=8iJMOBcPQyg