The 15th Amendment was ratified on February 3, 1870, giving men of color and former slaves the right to vote. Women generally did not have the right to vote until passage of the 19th Amendment in 1920. Ratification of the 15th Amendment was seen as the last piece to resolving African-American rights resulting from the ending of slavery with the ratification of the 13th Amendment and the granting of rights of citizenship under the 14th Amendment.
As simple as the amendment is in words, it was not so simple to implement. Impediments were instituted, particularly by Southern states, to block voting by African-Americans and others of color, including poll taxes and literacy tests. Those issues were addressed almost 100 years later with ratification of the 24th Amendment in 1964 and the passage of the Voting Rights Act of 1965.
To read more about the 15th Amendment, see:
Unintended Consequences of Constitutional Amendment, ed. by David E. Kyvig.
Champion of Civil Rights: Judge John Minor Wisdom, by Joel William Friedman.