Article VI, paragraph 2, known as the "Supremacy Clause," states that the Constitution itself and the laws of Congress supersede state and local laws. Federal supremacy is the first principle way that the US Constitution creates a stronger national government than what had existed under the Articles of Confederation.
In addition to the Supremacy Clause, the superlative nature of federal power is both outline and circumscribed within other Constitutional provisions. Articles I–III enumerate exclusive powers for each branch of government. In addition to naming itself the arbiter of what is and is not constitutional in Marbury v. Madison, the Supreme Court has consistently upheld this power. Federal power is not, however, unlimited: states retain sovereignty over matters strictly within their borders that are not the purview of the federal government. The 10th Amendment specifically guarantees this.
On the balance, the federal government created by the Constitution was much stronger and could much more effectively manage interstate disputes as well as the affairs of the nation as a whole than the Continental Congress had been. For example, under the Articles of Confederation, Congress could not raise taxes on its own but had to ask for funding from the states, who could refuse. Congress also had the power to declare war but was dependent on the states for troops. Finally, the "President" under the Articles was merely the President of the Continental Congress and had little power. There was no chief executive, which made swift and muscular exercises of power difficult.
Thus, the second major way the Constitution allowed for a stronger national government was in the creation of the Presidency. The Articles were written at a time when Americans were fighting for independence and strongly feared tyranny. After the war was over, however, it became apparent that a powerful chief executive would be needed to prevent further uprisings like Shay's Rebellion. When Daniel Shays and his men marched on the Springfield armory with the intent of overthrowing the government, Congress could not act quickly or effectively, and it came down to the Massachusetts Militia to quell the uprising.
Spurred on by this crisis, the Framers sought to create a position for George Washington that would give him many of the powers of a monarch—but one who was elected and subject to limits like impeachment. With this power, Washington and his successors could move quickly, like a Roman dictator, to crush future rebellions without having to wait for Congress.
We don't always notice the powers of a President in times of peace. While the President executes the laws of the United States, that power in reality is typically carried out in near-autonomy by the departments of the executive branch such as the Department of State, Treasury, or the Interior. Similarly, while the President heads the armed forces, the day-to-day decisions of the military are typically left to the command structure of the armed forces themselves. The President typically asserts power, rather, by deciding who heads these various organizations and by guiding the general direction of their policies.
It is in times of emergency or in matters of the greatest national importance that having a President is most useful. During a national disaster or foreign attack, the President can order immediate response. On 9/11, President Bush closed our national airspace to prevent any further damage from hijacked airliners. The President is also our head of state. He/she, in effect, represents us to the world and to other heads of state. The personal diplomacy between FDR and Winston Churchill during WWII allowed for a very close level of cooperation between the US and Great Britain in military and diplomatic matters.