Reconstruction refers to the period 1865-1877, immediately after the end of the Civil War. Eleven Southern States had seceded. At the end of the war all but Tennessee were under military occupation and martial law (Tennessee had been largely occupied since 1862 and thus escaped postwar reconstruction). Reconstruction was the process by which the seceded states were gradually allowed to re-enter the American political process under new, Union-loyal leadership. The other goal was to integrate the newly freed slaves into American life.
It was largely seen as a failure. White Southerners refused to allow blacks their rights of citizenship, and after a while white Northerners got tired of forcing the issue. It took another century, and the 1960s Civil Rights movement, for blacks to have full rights in the South.