Clause 1. The Congress shall have Power To lay and collect Taxes, Duties, Imposts and Excises, to pay the Debts and provide for the common Defence and general Welfare of the United States; but all Duties, Imposts and Excises shall be uniform throughout the United States;
The very first power given to Congress by the Constitution is the power to tax. Money is power, and in the governmental structure created by the Constitution, Congress—not the president—controls the money. Congress also has the power to levy tariffs (taxes on imported goods) but it's not allowed to charge more for imports into one state than into another. The Framers of the Constitution probably put the tax power first on the list of Congress's enumerated powers because they were acutely aware that one of the biggest problems of the old Articles of Confederation was that its version of Congress did not have the power to tax, and thus didn't have the power to do much of anything at all.
Clause 2. To borrow Money on the credit of the United States;
Congress is allowed to go into debt to pay for government programs and services. Deficit spending by the government was fairly rare in peacetime through much of American history, but has been quite common in recent decades.
Clause 3. To regulate Commerce with foreign Nations, and among the several States, and with the Indian Tribes;
Congress has the power to impose regulations on interstate and international business. This "interstate commerce clause" has been quite controversial in the history of constitutional law; for a long time, judges tended to read the clause narrowly, overturning federal laws they deemed focused mainly on regulating economic activity within states rather than between them. Since the 1930s, however, judges have tended to read the clause broadly, allowing the government to regulate all kinds of economic activity—by setting a national minimum wage, for example.
Clause 4. To establish an uniform Rule of Naturalization, and uniform Laws on the subject of Bankruptcies throughout the United States;
Another clause that seems to bring together two things that have little in common. First, Congress has the power to set up a process for immigrants to become American citizens. (The idea that America is a "nation of immigrants" is thus embedded right in the Constitution.) Second, Congress has the power to set rules for hopelessly indebted people and businesses to declare bankruptcy. In 2005, Congress used that power to change bankruptcy law; it's now much harder for individuals to escape credit card debts by declaring bankruptcy.
Clause 5. To coin Money, regulate the Value thereof, and of foreign Coin, and fix the Standard of Weights and Measures;
Congress controls the minting of money and (theoretically) sets its value. In practice, the creation of the Federal Reserve Bank in 1913 transferred most of the power over setting the value of the dollar to the Fed. Congress also gets to set standards of weights and measures; in the 1970s, this became controversial, as traditionalists in Congress blocked President Jimmy Carter's attempts to begin a switchover to the metric system.
Clause 6. To provide for the Punishment of counterfeiting the Securities and current Coin of the United States;
Printers of funny money beware!
Clause 7.To establish Post Offices and post Roads;
For most of the first century of American independence, the Post Office was by far the largest and most important organization within the federal government. Congress has the power to set up Post Offices and to build roads connecting them.
Clause 8. To promote the Progress of Science and useful Arts, by securing for limited Times to Authors and Inventors the exclusive Right to their respective Writings and Discoveries;
Congress has the power to set up a system of copyrights and patents, granting creative people the exclusive right to sell their creations.
Clause 9. To constitute Tribunals inferior to the supreme Court;
This means that Congress has the power to set up lower-level federal courts that report to the Supreme Court. That court system has grown over time; today there are twelve circuit Courts of Appeals, plus 94 federal District Courts, plus dozens of other special courts.
Clause 10. To define and punish Piracies and Felonies committed on the high Seas, and Offences against the Law of Nations;
Congress has the power to punish pirates. Amazingly, after a period of hundreds of years when piracy seemed to be a thing of the past, in 2009 piracy once again became a hot topic when Somali pirates began targeting merchant ships off the Horn of Africa.
Clause 11. To declare War, grant Letters of Marque and Reprisal, and make Rules concerning Captures on Land and Water;
This clause grants Congress one of its most important powers: the power to declare war. Congress, and only Congress, can officially do so. (The President can't!) This clause also grants Congress one of its more bizarre powers: the power to hire pirates to attack the nation's enemies. (That's what a "Letter of Marque" is... a letter that gives a pirate official permission to do his thing in the name of the national interest. Avast, ye mateys!)
Clause 12. To raise and support Armies, but no Appropriation of Money to that Use shall be for a longer Term than two Years;
The Founding Fathers were really worried about the danger of standing armies, the kind of permanent professional armed forces that had, they felt, been used by the British monarchy to oppress them before the Revolution. So they carefully divided the power to control the military between the executive and legislative branches; the president is Commander-in-Chief but only Congress has the authority to pay (or not pay) for military actions. Further, Congress cannot fund military operations more than two years in the future.
Clause 13. To provide and maintain a Navy;
This one's pretty self-explanatory.
Clause 14. To make Rules for the Government and Regulation of the land and naval Forces;
Congress has the power to set rules for the behavior of the armed forces. From 1806-1951, those rules were contained in a law called the Articles of War. Since 1951, they have been contained within the Uniform Code of Military Justice. Any soldiers or sailors who violate those rules face court-martial.
Clause 15. To provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions;
Congress has the power to call out the militia—organized units of citizen soldiers—to defend the nation from attack or armed rebellion. In modern times, the militia has been replaced by the National Guard.
Clause 16. To provide for organizing, arming, and disciplining, the Militia, and for governing such Part of them as may be employed in the Service of the United States, reserving to the States respectively, the Appointment of the Officers, and the Authority of training the Militia according to the discipline prescribed by Congress;
Control over the militia is divided between Congress and the state governments. If the militia is called into national service, Congress pays for it and governs its actions. The states, however, retain control over who serves as its officers and how its men are trained. These distinctions were probably more important in the 1790s than they are today.
Clause 17. To exercise exclusive Legislation in all Cases whatsoever, over such District (not exceeding ten Miles square) as may, by Cession of particular States, and the Acceptance of Congress, become the Seat of the Government of the United States, and to exercise like Authority over all Places purchased by the Consent of the Legislature of the State in which the Same shall be, for the Erection of Forts, Magazines, Arsenals, dock-Yards, and other needful Buildings;
Congress has the power to set up a national capital of the United States that is outside the jurisdiction of any state. (Congress used this power to create Washington, DC, on swampland along the Potomac River that was originally part of Maryland.) Congress also has ultimate authority over all federal military facilities, even if they're located within particular states. And that "--And" means were getting almost to the end of this long list of Congress's enumerated powers.
Clause 18. To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.
This, the so-called "elastic clause," is the basis for all of the legislative branch's implied powers (powers not explicitly listed in the Constitution but held to be legitimate because they are "necessary and proper" for the Congress to exercise the other powers that are listed here. Over time, this clause has been used to justify a gradual expansion in the general power of Congress and the entire federal government.