Woman suffrage, or the legal right of women to vote, was the result of a movement which began in the 1840s. Women's right to vote was established first in various states and localities, sometimes on a limited basis, and then nationally in 1920. The Nineteenth Amendment, which prohibits states and the federal government from restricting voting on the basis of sex, was ratified on August 18, 1920.
For further reading:
Women’s Suffrage: The Movement, Social Welfare History Project
Frederick Douglass on Woman Suffrage: 1888, Social Welfare History Project