What Civil War issues remained unresolved by the twentieth century?

Institutional and individual racism were absolutely (and still are) issues from the Civil War that were left unresolved by the turn of the twentieth century. The few gains that were won through the Reconstruction Era were quickly rolled back due to immense vigilante racism that coursed through the South. Black folks and anti-racist whites who were involved in reconstruction work were consistently attacked and murdered throughout the South as the KKK and other racist groups began to form and prevent racial progress. Some of these groups then gained the support of President Andrew Johnson and local police and political organizations. Through this violence, Jim Crow laws were created that essentially criminalized black existence. While the abolition of slavery through the 13th Amendment is generally understood to have eradicated slavery in the United States, the 13th Amendment specifically states that slavery is legal and acceptable in the form of punishment for a crime (incarceration). As such, prisons were created across the South and filled with recently freed black people as Jim Crow laws criminalized most of black existence. These prisons then sold the labor of their prison population to private individuals, companies, and state organizations. As such, much of the problems of the Civil War were left unresolved or morphed into different forms of the same type of oppression.

Arguably the most significant legacy of the Civil War at that time was the issue of racial equality. At the turn of the century, millions of African Americans in the South were systematically denied basic civil rights, such as the right to vote and the right to stand for public office. The Civil War had been fought to end slavery, yet despite the Union victory and the subsequent passing of the Fourteenth Amendment, Southern states had been allowed to construct an entire legal edifice of discrimination that effectively reintroduced slavery by the back door, in substance if not in form.

The so-called Jim Crow laws were buttressed by political indifference at the federal government level. To put it bluntly, there simply weren't many votes for civil rights from either party. In addition, the US Supreme Court refused to dismantle Jim Crow, ruling in the notorious case of Plessy v. Ferguson (1896) that racially segregated facilities were constitutional so long as they were "separate but equal."

What issues from the Civil War period still remain?

There are two main issues that remain from the Civil War period.

First, there is the basic political issue that caused the Civil War.  This is the issue of states’ rights and federalism.  The issue is nowhere near as salient today as it was at the time of the Civil War, but it does still remain with us.  Many conservatives believe that the states need to be given back more of the powers they once had.  They believe that the federal government has become too strong.  This is part, for example, of the reason why Republicans are so deeply opposed to “Obamacare.”

Second, there is the issue of African American rights.  Of course, we are no longer fighting over slavery, but there are still issues about the rights of African Americans.  We still argue about what, if anything, needs to be done to make up for the disadvantages that have come to African Americans because their ancestors were enslaved and then discriminated against.  This is an issue that is closely related to, though not exactly the same as, the issue of slavery that was so important at the time of the Civil War.