Independence Day in Mexico is September 16. People gather in the zocalo (town center) on the 15th, and at midnight the president comes out on the balcony, remembering the leaders in the fight for freedom. He gives the "grito" about 4 minutes in, it's neat, you should stick around to see it. Someday I'd love to be there.
The song you hear in the beginning and after the grito is the national anthem. When people say that the anthem for the U.S. is too bloody, I always think of the Mexican himno nacional. Way bloodier. Here it is, with lyrics, if you're interested (English translation).
¡Feliz día de independencia!