The Gist: A slave was taken by his owner to Illinois, a free state. When the owner died, the slave sued the owner's estate administrator, claiming the slave's residence in a free state made him a free man. The Supreme Court held that slaves were property not citizens, and because they were not citizens, had no standing to sue in federal court (i.e. the court had no jurisdiction to hear the case).
In the same decision, the Supreme Court also declared the Missouri Compromise unconstitutional, effectively holding that Congress could not grant citizenship to slaves or their descendants because such action would be taking property from slave owners without due process or just compensation.
The Details: In 1803, The United States purchased the Louisiana territory (an extremely large chunk of today's America) from France and a national debate arose regarding what parts of the territory would be free states and slave states. In the Missouri Compromise, Congress determined Missouri as a slave state, prohibited slavery in the northern part of the territory and left it up to states in the southern party of the territory whether they wanted to allow slavery or not.
At the same time, Dred Scott was a slave in Missouri owned by John Emerson. Emerson eventually took Scott to Illiniois, a free state. When Emerson died, Scott sued John Sandford, Emerson's estate administrator. Scott's claim was that because he was now a resident in a free state, the death of his owner rendered him a free man. The Constitution allows a citizen of one state to sue a citizen of another state (Sandford was from New York) in federal court. This is known as diversity jurisdiction and still exists to this day.
The Supreme Court held that Scott did not have standing to sue because,as a slave, he was not a citizen of the United States--he was property. Because he was not a citizen, the court did not have diversity jurisdiction to hear the case. The decision rendered all slaves forever and permanently bound to their owners, unless emancipated.
Since the court determined Scott did not have standing to sue and the court did not have jurisdiction to hear the case, it could have (and should have) stopped there, but didn't. The court then declared the Missouri Compromise unconstitutional, stating Congress could not grant citizenship to slaves (by declaring certain territories free) because it deprived citizens of their property (i.e. slaves) without due process or compensation.
In one single decision, the Supreme Court officially declared slaves and their descendants forever property of their owners, without citizenship, without the right to sue in federal court and prohibited Congress from doing anything about it. It has since been declared one of the worst--if not the worst--and most racist decisions in U.S. Supreme Court history.
Dred Scott did far from resolve the issue of slavery, and if anything only made things worse, helping spark the fuel to the fire that became the Civil War. Ironically, following the decision,the descendants of Scott's first slave owner purchased emancipation for Scott and his family, rendering him a free man.