Mexican War of Independence
Although Mexico gained its independence in September 1821, the marking of this historical event did not take hold immediately. The choice of date to celebrate was problematic, because Iturbide, who achieved independence from Spain, was rapidly created Emperor of Mexico. His short-lived reign from 1821–22 ended when he was forced by the military to abdicate. This was a rocky start for the new nation, which made celebrating independence on the anniversary of Iturbide's Army of the Three Guarantees marching into Mexico City in triumph a less than perfect day for those who had opposed him. Celebrations of independence during his reign were marked on September 27. Following his ouster, there were calls to commemorate Mexican independence along the lines that the United States celebrated in grand style its Independence Day on July 4. The creation of a committee of powerful men to mark independence celebrations, the "Junta Patriótica", organized celebrations of both September 16, to commemorate Hidalgo's grito and the start of the independence insurgency, and September 27, to celebrate actual political independence.
Puebla
Cinco de Mayo—or the fifth of May—is a holiday that celebrates the date of the Mexican army's 1862 victory over France at the Battle of Puebla during the Franco-Mexican War (1861-1867). A relatively minor holiday in Mexico, in the United States Cinco de Mayo has evolved into a commemoration of Mexican culture and heritage, particularly in areas with large Mexican-American populations. 
Many Americans mistakenly believe that Cinco de Mayo (“May 5th”) is the Mexican equivalent of the United States’ Fourth of July holiday — a date marking the official casting off of colonial rule via the announcement of a new independent country. However, the Mexican version of Independence Day is celebrated on September 16, for it was on that date in 1810 that the commencement of the war for Mexican independence from Spanish rule was pronounced in the small town of Dolores by Miguel Hidalgo y Costilla (an event now referred to as the “Grito de Dolores” — “Cry of Dolores” — or “El Grito de la Independencia”).