Which group received the right to vote in 1870 when the 15th Amendment was ratified?
African-Americans were given the right to vote.
The answer is a bit more complicated than this, but that would be the one sentence version.
Here's the main part of the 15th Amendment, Section I:
"The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any state on account of race, color, or previous condition of servitude."
This Amendment, like the 13th and 14th, are coming after the end of the Civil War in 1865, and during Reconstruction. It is trying to take steps towards moving the country away from slavery.
The reason, though, the answer is more complicated than just saying that African-Americans were given the right to vote is that the Amendment only says that citizens shall not be denied based on their race, color, or "previous condition of servitude", which is a reference to people who were enslaved. (Who were, of course, people of African descent.)
Here's the problem: states are really the ones to decide who gets the right to vote. Many states in the South began making other qualifications to vote, which didn't say that Blacks/African-Americans couldn't vote, but were clearly aimed at this group of citizens.
Thus, many African-Americans remained disenfranchised for almost another century, which is why the Voting Rights Act was necessary. From history.com : "The Voting Rights Act, signed into law by President Lyndon Johnson (1908-73) on August 6, 1965, aimed to overcome legal barriers at the state and local levels that prevented African Americans from exercising their right to vote under the 15th Amendment (1870) to the Constitution of the United States. The act significantly widened the franchise and is considered among the most far-reaching pieces of civil rights legislation in U.S. history."