13th amendment: ratified in 1865, Abolished Slavery
"neither slavery nor involuntary servitude, except as punishment for crime whereof the party shall have been duly convicted, shall exist within the United States, or any place subject to their jurisdiction."
Banning slavery was constitutionally necessary because of the Emancipation Proclamation made by Abraham Lincoln in 1863, where the Southern States had to agree to give up slavery in order to come back into the Union.
15th amendment: ratified 1870, Right of Suffrage(The Right to Vote)
"The right of citizens of the United States to vote shall not be denied or abridged by the United States or any State on account of race, color, or previous condition of servitude."
This amendment only allowed men to vote. States did find ways to prevent blacks from voting, such as with the Jim Crow Laws and the KKK. Many blacks died during this time period to excercise their right to vote.
Amendments 13 and 15 are part of the Reconstruction Amendments.