The Bill of Rights, and the individual protections it ensures, has been almost universally extended to the State governments by a series of Supreme Court rulings. All of these rulings are drawn from the Court’s interpretation of the Fourteenth Amendment. The Amendment states, among other things, that state governments have no right to deny life, liberty, or property outside of the law of the land. This is called the due process clause.

The Seventeenth Amendment, passed in 1913, established that senators would no longer be chosen by a vote in the State Legislature, but would instead be elected by a vote of the people who lived in each senator's district. This represented the culmination of a decades-long struggle by the Progressives to extend greater suffrage rights and responsibilities to the people of the United States.

The passage of the 14th Amendment and the due process clause was interpreted by the Supreme Court to mean all persons have a right to equal due process, both before the federal and state governments. The federal government is supreme over state governments, and thus the states were forced to incorporate the Bill of Rights.

The Fourth Amendment is one of the most important parts of the Bill of Rights. It protects citizens from random government raids on their homes, and requires police to have probable cause or a court-sanctioned warrant to search a person's home. The Fourth Amendment is the reason the police, unless in an extenuating circumstance in which harm is imminent, need warrants to search homes and persons.

This answer requires relatively little explanation—it’s one of those questions that requires rote memorization. That said, there is a trick to remembering which amendment lifted prohibition (I believe we covered it in a previous question); 21 to drink—the 21st Amendment lifted prohibition.

As for the others, they are incorrect:

18th: Enacted Prohibition

19th: Granted the franchise to all women

20th: Does a number of things, primarily ending lame duck, and setting succession

23rd: Extends the franchise to residents of Washington DC (why was this necessary?)

The 15^th Amendment expanded the right to vote to people regardless of their race. The 19^th Amendment expanded the right to women. The 9^th Amendment reserves some unlisted powers in the Constitution for the states.  

The 14^th Amendment was ratified in 1868 and contains various provisions, including ensuring that the rights of all would be protected. This Amendment was focused on providing for the rights of the former slaves who faced discrimination and attempts to stifle their rights. This Amendment states that the government shall not “deny to any person within its jurisdiction the equal protection of the laws.” This part of the Amendment is referred to as the Equal Protection Clause and is often used to justify intervention by the federal government on behalf of those who are being discriminated against.

The Amendment was ratified to ensure the rights of former slaves and the birthright citizenship section was included to ensure that all former slaves are citizens and are entitled to have the same rights as other citizens. While slavery was allowed slaves were considered property, not citizens, so this amendment was needed to clear up confusion about the status of the former slaves and to ensure that their rights were protected.