The War Powers Resolution was a resolution of Congress in 1973 designed to win back control of the warmaking power from the Presidency. The Presidency had since World War II come to have a virtual monopoly over this power. As the rise of the modern Presidency coincided with the rise of the United States as a superpower, this gave the man in the White House a great deal of ability to impact the world stage.
Dissent from this trend had been going on for decades, but the Vietnam War really changed things, as it changed everything in American politics. The authority to go to war had been given in the Gulf of Tonkin Resolution, passed hastily by Congress following the alleged attack of American ships in the Gulf of Tonkin. Although this Resolution became the basis for the eventual introduction of 500,000 troops into South Vietnam, when it was passed few in Congress expected such an extreme result.
Many hoped that by taking a strong stand against the Communists they would discourage further aggression. Of course, in so thinking they had drastically underestimated the character of the North Vietnamese regime, which would never rest until it had reunified the country by force. And because successive American Presidents were so keen to stop this from happening, the troops just kept on going. Eventually in 1973 the Congress legislated to make illegal any military action 'in or over or off the shores of' Indochina. This in itself was a decisive challenge to the executive, but in theory the War Powers Resolution was much more revolutionary.
The idea of the War Powers Resolution is that if American forces are going to be sent into combat, or somewhere where they're likely to enter combat, or even if the peacetime military presence in a country is being substantially increased, then Congress has to have some say on the matter. In theory, this has two benefits.
Firstly, it stops actions like the 1973 ban on military action in Indochina being necessary. Foreign policy cannot be made by case-specific, inflexible legislation. Hence, when it became necessary to evacuate Americans and friendly Indochinese from Da Nang, Phnom Penh and Saigon in 1975, it was technically illegal to use American military forces to assist. In the end, President Ford cited his authority (and indeed obligation) as Commander in Chief to protect American citizens. Of course, the question of the Vietnamese and Cambodians also rescued is a different matter.
The conflict between American lives and the 1973 ban on combat in Indochina would never have come about if not for a breakdown in trust between the legislature and the executive. By trying to ensure that there was collective agreement on military deployments abroad from the start, the War Powers Resolution sought to stop such case-specific legislation being necessary.
The second theoretical benefit is even more contentious. It is contended that the Resolution makes the warmaking power more democratic, and more in line with what was intended by the founders of the Republic. By allowing the Congress to decide on matters of war and peace, the right decision is more likely to be reached rather than if the executive can do what it wants. Of course, the problem here is that the Congress doesn't necessarily have great expertise in foreign affairs either.
Even worse, it has proved itself repeatedly incapable of following a path and sticking to it for a long time. Consistency is required for a successful foreign policy, and there is no God-given assurance that wars which are necessary for the national interest will be popular. However, the alternative of letting the executive decide entirely on its own about matters of war and peace can lead to substantial alienation between the executive and the rest of political society.
Since the Resolution was passed, Congress has never really been consulted by a President about war (the Resolution is anyway highly vague about how such consultation should take place, or even who exactly in Congress should be consulted). Usually the executive informs Congress after the fact what has happened (in the case of small events), or receives authorization through a joint resolution of both houses of Congress. For instance, Operation Iraqi Freedom was authorized by the "Joint Resolution to Authorize the Use of United States Armed Forces Against Iraq" on 11 October 2002.
A final point worth considering is the constitutionality of the War Powers Resolution. The war powers clause of the Constitution (Article One, Section Eight, Clause Eleven) reserves for the Congress the right to make war, but this has only been invoked five times in America's history. The last such instance was World War II, following which Presidents have preferred to rely upon their authority as Commander in Chief to commit troops abroad. The Supreme Court has never ruled on this matter, and most likely never will until the day a President requests a war resolution and none is forthcoming.