During the time of Reconstruction (after General Lee surrendered to the Union army, after Lincoln was assassinated), there was a lot of debate as to how to let the southern Confederate states back into the Union. Some people wanted it done quickly in order to move on with life. However, some wanted the southern states to apologize for leaving the Union. This group wanted a more formal apology and wanted to require people living in the south to make an official oath of allegiance to the Union. Look up the Ten Percent Plan and Wade-Davis Bill. Tennessee was the first state to get back into the Union (President Johnson was from Tennessee).
President Andrew Johnson (who took over in April 1865) was on the side of getting back to normal as quickly ass possible. The Radical Republicans did not agree with him. They were able to take the majority of Congress in 1866 and block Johnson's legislative acts so he couldn't do the things he wanted to do. The term "impeachment" is important. Johnson was the first President ever to be impeached by the House of Representatives. Impeachment is when an elected official (like the President) can be removed from the office they hold. The Senate saved Johnson by 1 vote.
Ulysses S. Grant was the next President to take office. He was a Radical Republican and believed in more rights for African Americans. The Ku Klux Klan was created during this time and tried to physically hurt African Americans but Grant used his power to suppress this.
The 13th, 14th, and 15th Amendments were passed and this is also very important. 13th - abolished slavery; 14th - Anyone born in the United States is automatically a citizen and has all privileges of citizenship (life, liberty, pursuit of happiness, due process, etc); 15th - the government cannot deny a citizen the right to vote based on race (note: women could still not vote).
I would also look up the Compromise of 1877 which most people consider the end of the Reconstruction Era. Basically, it ordered federal troops (military) out of the Southern states.
Hope this helps!