What was the significance of Mapp v. Ohio?
The significance of this case was that it introduced what is called the "exclusionary rule" to the legal systems of the American states. (It had already existed on the federal level.) This is the rule that says that illegally obtained evidence (gotten without a warrant, more or less) cannot be used against a defendant in court.
The 4th Amendment protects us against illegal searches and seizures. So let's say the police illegally search your house and find evidence of crime. What can you do? Before Mapp, all you could do was complain. After Mapp, you (if you had a decent lawyer) could get that evidence excluded -- the prosecution could not use it against you in any way.
The significance is that it is part of a line of cases that enforce Bill of Rights protections to state and local governments. Specifically, the exclusionary rule of the Fourth Amendment must be followed by state and local enforcement agencies through the due process clause of the 14th Amendment. The 14th Amendment acts as a limitation on the states.
Back in the early 1800s a man sued the City of Baltimore claiming that by its actions in diverting the flows of water into baltimore harbor, the city had effectively taken his wharf without just compensation as required by the 5th Amendment. The Supreme Court through Chief Justice Marshall said that the 5th Amendment only protected him from the federal government not from the City of Baltimore. (Barron v. Baltimore)
When the 14th Amendment passed, one of its intended purposes was to incorporate the first eight Amendments as limitations upon the states. That didn't happen however. See, the Slaughter-House cases. But what the Supreme Court began to do on a case-by-case basis was incorporate specific Bill of Rights protections to the states through the 14th Amendment's due process clause. So for example in 1925 the Court in Gitlow v. New York extended the 1st Amendment freedoms of speech and the press to the states. Just this past year the Supreme Court extended the 2nd Amendment right to bear arms to the states in a case called McDonald v. Chicago. There are a few bill of rights protections that are not incorporated such as the right to indictment by grand jury and the right to bail to name just two.