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The most important event was obviously the American Civil War from 1861-1865. This led to the 14th Amendment which defined a citizen as "any person born or naturalized in the US." The 13th Amendment abolished slavery, and the 15th Amendment prohibits denying the right to vote to someone because of their race.
Other key events include the Louisiana Purchase, the Texas Revolution, The Spanish-American War, the War of 1812, the purchase of Alaska, the Mexican Cession, the Indian Wars, and the California Gold Rush.
Like I said, the right to vote was extended to all male citizens by the 15th Amendment. Women were not given the right to vote until the 19th Amendment was passed in 1920.
Population in 1770 = 2,148,100
Population in 1870 = 38,558,371
The historical population estimates from Wikipedia: http://en.wikipedia.org/wiki/Demographic_history_of_the_United_States