Full Faith and Credit shall be given in each State to the public Acts, Records, and judicial Proceedings of every other State. And the Congress may by general Laws prescribe the Manner in which such Acts, Records and Proceedings shall be proved, and the Effect thereof.
Full text available at the US government transcript site.
the different sections of Article IV.
This article is all about the States of the Union. It ensures that each state respects the laws of the others and requires that Citizens be treated equally under the law across the entire country. It even goes so far as to enforce the republican form of government (sometimes called a representative democracy) and obligates the federal government to help defend the states against invasion. As always, I highly recomend reading the original language at the government archive site either before or after reading my explanation. It would be a good idea to read the Federalist and Anti-Federalist papers as well as James Madison's notes on the Constitutional Convention if you want to understand what the founding fathers were so worried about. Equal treatment of citizens is what it's all about!
"Full Faith and Credit..." Section one is really about the states recognizing the legal decisions of the other states. For example a couple of people get married in Florida and then move to Georgia, they are still recognized as married in Georgia since the states have to accept the legal decisions made in the original state. The same goes with driver licenses and contracts between people in one state or another. State courts are not allowed to refuse to hear a case based on a nearby states laws but also can't force a case to be held only in their own court if it involves a nearby state. If a person commits a crime in Alabama, is convicted, and then runs away to Ohio the courts and police in Ohio will send him back to Alabama to face the sentence for the crime.
"The Citizens of each State shall be entitled to all Privileges and Immunities of Citizens in the several States.". This means that every citizen of the states is treated in a fair and equal way. To go back to the example above, if a citizen of one state commits a crime and then runs away to a different state the state he ran to will send him or her back home to face a trial for the crime.
This section is all about new states being added to the United States. We originally started with 13 colonies that became states. In the over the 200 years since then we have grown to 50 states! The states are not allowed to split into smaller states or combine into larger states unless they get permission from Congress and the government of both states agree. This section also talks about federally owned lands and how they have to be taken care of and managed. Basically the Congress gets to decide what to do with land owned by the US government. They also get to decide on what rules and regulations will be used to take care of the land.
The United States shall guarantee to every State in this Union a Republican Form of Government, and shall protect each of them against Invasion; and on Application of the Legislature, or of the Executive (when the Legislature cannot be convened), against domestic Violence. This really means that the US government is required to defend the states from outside threats (like if Cuba decided to invade Florida along with the Russians like the old "Red Dawn" movie). The US government has to help defend a state that is having internal problems as well but only if the state Legislature asks for help. This really is tied into the guarantee of a "republican form of government" so if the state is suffering from an internal attack that threatens to destroy their internal government (or turn it away from a representative democracy and into a dictatorship) they can ask for help from the federal government.