13th Amendment
The 13th amendment abolishes slavery permanently in the United States. Aside from wanting this in the Constitution, an amendment was necessary because Abraham Lincoln's Emancipation Proclamation only applied to states in rebellion. There were four slave states that fought on the Union side.
14th Amendment
The 14th amendment declares that everyone born in the United States is a citizen, including former slaves. Also, it states that everyone is entitled to equal treatment under the law by both the federal and state governments. After the Civil War, slaves were free in the South, but the state and local governments still treated them unequally. This amendment tried to make clear that such actions were illegal and that all levels of government had to respect the freedoms in the Bill of Rights.
15th Amendment
The 15th amendment states that race cannot be a factor in denying someone the right to vote. Unfortunately, the South got very creative in designing voter laws that seemingly treated everyone of all races equally. In reality it only applied to African Americans such as a passing a literacy test or paying a poll tax.