The whole thing is part of the schizoid personality of the University of California - it's technically one university (all the degrees read "The Regents of the University of California, upon the nomination of the faculty at X") but each campus has its own personality and sports teams - all the colors are blue and gold, but different shades of blue are used. Only UC Berkely is known as "Cal" - the rest are known by the abbrviations of their cities (UCLA, UCSB, UCSC, UCSD, etc) or just by the city name (e.g. "Davis" "Riverside" Irvine").
"Big C" is unquestionably the most famous and controversial Cal song. "Big C" was composed in 1913 by Harold P. Williams, with words by Norman Loyall McLaren. It was written to commemorate the creation of the large cement "C" built on the "rugged Eastern foothills" of the Berkeley campus in 1905, and the song was later entered in the Daily Californian's then annual school song competition. In the Fall of 1913, the competition was stiff, but the Rally Committee managed to narrow the field down to two songs, "Big C" and "Stanford Jonah." "Big C" took the prize with "Jonah" winning the next year.
The controversy surrounding the song has its roots in the "All University Weekend," an annual event which began around 1948 and lasted into the 1960's. This event was a double header football game that pitted Cal against UCLA and UC Davis against UC Santa Barbara. The games were played alternately in Berkeley one year and in Los Angeles the next year. Bands from all four of the schools would perform together in one giant, combined half-time show.
In one of the last "All U Weekends," F. Kelley James, then Associate Director of the UCLA Band and alumnus of the Cal Band wrote an arrangement of "Big C" for the combined half-time show. Afterwards, UCLA continued using his arrangement of "Big C," adding its own lyrics and renaming it "Sons of Westwood." The UCLA Band began playing it regularly as their new fight song. James Berdahl, then director of the Cal Band, was incensed over what he felt was a violation of the sanctity of Cal songs. A bitter exchange ensued between Berdahl and James for the next several years concerning the legal and ethical grounds under which "Big C" was appropriated. The matter came to a head in February 18, 1969, when Irwin Coster, working on behalf of the UCLA cause, received official word from the Copyright Office of the Library of Congress that "Big C" had never been copyrighted, and thus it was in the Public Domain. Public Domain status meant that only adaptations and arrangements of the song could be copyrighted, so UCLA had every legal right to "steal" the song. Some regents and UCLA administrators thought it quite reasonable that this "little sister" of Cal maintain "Sons of Westwood" as a "reaffirmation [of the University of California's] solidarity." However, ardent students and alumni at Cal were never happy with the situation, especially Berdahl, who continued to fight for the abolition of "Sons of Westwood" through the remainder of his tenure as director. Ironically, the nation at large recognizes "Sons of Westwood" as UCLA's fight song rather than Berkeley's as a result of their successful sports programs and extensive exposure on televised games, but that may be changing with the success of Cal Sports in the 21st century!