Perhaps the most important changes to occur as a result of the Civil War was the passage of the Thirteenth, Fourteenth and Fifteenth Amendments to the Constitution. The Thirteenth, passed while the war was still raging, ended slavery. The Fourteenth, perhaps the most far reaching amendment to the Constitution, gave citizenship to former slaves and guaranteed them the equal protection of the law:
All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.
The broad language of the Fourteenth Amendment has been construed by the Supreme Court to extend most (but not all) of the protections of the Bill of Rights against infringement by the individual states. Originally, the Bill of Rights only protected against abuses of the Federal government. The famous decision of Miranda vs. Arizona ("you have the right to remain silent...") was only possible because of the Fourteenth Amendment.
The Fifteenth Amendment guaranteed citizens of the U.S., particularly former slaves the right to vote:
The right of citizens of the United States to vote shall not be denied or abridged by the United States or by any State on account of race, color, or previous condition of servitude.
It is highly unlikely that these amendments would have passed were the South in a position to block them, as it was prior to the war. The effect of these amendments is still of vital importance.
In addition to the above, while the South had no representation in Congress, Congress passed laws which provided for the Transcontinental Railroad which took a northerly route. Again, this probably would not have been possible had the South remained active in Congress.
Of course, the most important change in the US as a result of the Civil War was the abolition of slavery. This allowed the country to, eventually, become a much more just society in racial terms.
The Civil War also helped to facilitate westward expansion and the settling of the plains states and the west. With the Southerners out of Congress, it was much easier to pass laws like the Homestead Act and the laws creating the transcontinental railroad. These sorts of laws helped bring about the movement of settlers into the parts of the country that were still unsettled by white people at that point.
In these ways, the Civil War both ended slavery and helped to create the westward expansion that was so important in US history.