What was the name given to the period of the time in which the united state struggled with how to rebuild the south and bring back in to the union

The answer is the Reconstruction Era.

For the beginning of the Reconstruction Era either 1863, which was the year that slavery legally ended, or the 1865, which is the end of Confederacy and it lasted until 1897. Reconstruction for a goal had the end of slavery and to end the nationalism of Confederacy.