Essentially, Congress made certain that Southern states followed the laws with regard to Reconstruction by force. Congress sent the military to the South to rule the Southern states and ensure that they obeyed the law.
Under the Military Reconstruction Act, the South (with the exception of Tennessee) was split into five military districts. Each of these districts was commanded by an army officer who had the right to do whatever was needed to keep order and protect people's (including blacks') rights. They were even authorized to create military tribunals instead of having regular civil courts in which to try people charged with crimes.
These military districts were disbanded after states complied with the rules and came to be allowed to have democratic governments again. However, military forces remained in the South in some areas.
One of the most significant moves by Congress to ensure the Southern states followed the law was the passing of the Reconstruction Act of 1867. The act ensured that Southern states’ constitutions were developed in a participatory manner by both blacks and whites getting involved through the principle of Universal Male Suffrage. The Act also required that all states recognized anyone born or naturalized in the U.S. as a citizen by ratifying the 14th Amendment in order to join the Union. The passing and subsequent ratification of the 15th Amendment by the states also ensured that everyone had a right to vote regardless of their race.
President Johnson who attempted to actively oppose the Reconstruction Act was impeached. He narrowly avoided conviction and was forced to sit out the rest of his term. Enforcement Acts were passed outlawing organizations that were engaged in racial violence. These laws allowed the President to declare martial law in areas under threat from such organizations. The federal government subsequently exercised its power by cracking down on such organizations and showing its commitment to Reconstruction.