Clause 1. The Migration or Importation of such Persons as any of the States now existing shall think proper to admit, shall not be prohibited by the Congress prior to the Year one thousand eight hundred and eight, but a Tax or duty may be imposed on such Importation, not exceeding ten dollars for each Person.
This is another euphemistic nod to America's dark history of slavery. "Such Persons as any of the States now existing shall think proper to admit" is a really long-winded way of saying "slaves" without actually saying "slaves." The Constitution barred any attempt to outlaw the slave trade before 1808. As soon as that date rolled around, Congress did vote to block the international slave trade, although slaves continued to be sold within the country and slavery itself lasted for almost another 60 years.
Clause 2. The Privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it.
The Writ of Habeas Corpus is perhaps the most important foundation of civil liberties. "Habeas corpus" is a Latin term meaning "you shall have the body"; in practice, the right to Habeas corpus means that you can't be held in jail without facing legitimate charges of some kind—there is no such thing as indefinite detention without due legal process. Violations of habeas have been quite controversial in American history. During the Civil War, Abraham Lincoln (in one of his most criticized moves ever) suspended habeas; during the War on Terror, George W. Bush controversially argued that terrorism suspects held at Guantanamo Bay, Cuba, had no right to habeas and thus could be held indefinitely without trial.
Clause 3. No Bill of Attainder or ex post facto Law shall be passed.
A bill of attainder is a law that simply declares, by legislative fiat, that certain people are guilty of a crime and then imposes some kind of punishment upon them. In other words, it's a way for a legislature to act like judge and jury, convicting and punishing people without benefit of trial. Bills of attainder used to be used occasionally by the British Parliament; the American Founding Fathers viewed them as terrible violations of liberty and banned them from the United States.
An ex post facto law is a law that retroactively criminalizes a certain act after it has already been committed. In other words, it would allow a person to be prosecuted for doing something that wasn't actually illegal yet at the time they did it. The framers of the Constitution viewed ex post facto laws, like bills of attainder, as blatant abuses of power and banned them.
Clause 4. No Capitation,
or other direct, Taxshall be laid, unless in Proportion to the Census or Enumeration herein before directed to be taken.
A capitation tax is a "head tax," one charged to each individual in the population. This clause required Congress to levy any taxes on the basis of a state's population, not on the basis of individual income or any other standard. The 16th Amendment, passed in 1913, struck the reference to "other direct Tax[es]", making it possible to create the modern personal income tax system that we know and love today.
Clause 5. No Tax or Duty shall be laid on Articles exported from any State.
Southern economies at the time of the Constitutional Convention depended upon the export of cash crops like cotton, tobacco, rice, and indigo. Those states insisted that the Constitution ensure that those exports wouldn't be taxed by the national government. In the so-called Commerce Compromise, the northern states agreed. Congress does have the power to tax imports, though.
Clause 6. No Preference shall be given by any Regulation of Commerce or Revenue to the Ports of one State over those of another: nor shall Vessels bound to, or from, one State, be obliged to enter, clear or pay Duties in another.
This clause is designed to ensure that all states are treated fairly by the national government. Congress can't charge taxes for shipping goods from one state to another, and it can't favor one state's ports over another through preferential regulations or taxes.
Clause 7. No Money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of Receipts and Expenditures of all public Money shall be published from time to time.
This clause is critically important, granting Congress (and only Congress) the "power of the purse"—that is, control over government spending. The president can't get his hands on one dime of the public's money without Congress first approving that spending in an appropriations bill. Congress's control over the government's money is perhaps the most important check against unlimited presidential power.
Clause 8. No Title of Nobility shall be granted by the United States: And no Person holding any Office of Profit or Trust under them, shall, without the Consent of the Congress, accept of any present, Emolument, Office, or Title, of any kind whatever, from any King, Prince, or foreign State.
Here the Framers of the Constitution sought to ensure that the United States would never develop a formal aristocracy such as that which ruled Britain. The US government cannot grant any titles of nobility; here in America, we have no counts or dukes or earls. Further, no one working for the government is allowed to accept a grant of nobility from a foreign government, either. This was mainly included as an attempt to block foreign corruption of US government officials.