Chicano or chicana (feminine form) came into use during the 1960s for Mexican Americans. This is not surprising because this era was the heyday of the civil rights and anti-war movements. Mexican Americans shared many of the problems afflicting the black community of the United States: poverty, discrimination, crime, and exploitation by employers. The term "Chicano" was initially pejorative, but it came to be a badge of honor for Mexican Americans.

The Chicano Movement (El Movimiento) or "Brown pride" movement was strong circa 1966 to 1977. Cesar Chavez's efforts to organize Chicanos who worked on farms was one aspect of the movement. Chavez wanted to stop the exploitation of Chicanos whose labor produced grapes, lettuce, and other agricultural products. But the Chicano Movement was much broader than the fight for union rights for farm laborers. Chicana leaders were feminists who fought against both racial and gender discrimination. As a whole, the Chicano Movement was much less well-known than the civil rights movement. But it was significant—especially for Mexican Americans.

Today, ethnic studies departments at American universities offer some courses in Chicano Studies. These courses explore the social, historical, and cultural background of Mexican Americans.

There is no single accepted explanation of the term "Chicano."

My understanding of the origins of this term is taken from the Encyclopedia of Race and Racism, published by MacMillan Reference USA.  In the entry for the term "Aztlan," the encyclopedia argues that the term "Chicano" comes from the word "Mexicano."

It is a fact that the people that we refer to as "Aztecs" called themselves "Mexica."   This word was pronounced with the "x" having a "sh" sound.  This meant that the word "Mexicano" could be pronounced "Meshicano."  From there, the sound shift to "Mechicano" is a small one, and the term "Chicano" can easily be seen as an abbreviation of this word.

This is, of course, simply an informed supposition on the part of scholars.  There is no direct proof of the origin of the word.