The 14th and 15th amendments are considered the greatest achievements of Reconstruction because they gave a significant amount of rights to African-Americans.
The 14th amendment states that any person born on American soil is automatically a United States citizen. This law was enormous for individuals who used to be slaves, as many of them were born in the US and had no legal rights at all. This amendment gives them all the protections/rights guaranteed by the US Constitution.
The 15th amendment gives African-American men the right to vote. This is the first time in American history that African-Americans can vote on a national scale. This allows them to influence what laws are made and who runs our government.