The government is operated on the tenants of Federalism, which balances the powers between separate branches of government.
Here's a basic rundown of the US govt, which is composed of the Executive Branch (President, the Executive Office of the President, the Office of Management and Budget, federal bureaucracy and all the departments etc), the Legislative Branch (Congress) and the Judicial Branch (Federal and Supreme Courts).
The Constitution placed most of the powers in the legislative branch, which makes sense since they are the most representative of the people, and since the Constitution's designers had just witnessed an overthrowing of an absolute monarch. But the changing political environment has shifted the balance of power to the Executive.
Executive Branch:
The President signs or vetoes legislation, the ultimate check on Congress.
The OMB and the President create the blueprint for the national budget plans each year (for Congress to debate and sign), a power Congress gave to the Executive in the early 20th century.
The President can issue Executive Orders which are like treaties but which can be canceled by the Executive at any time.
The President nominates federal court judges and Supreme Court Justices; the Senate appoints or rejects them.
The President can call special sessions of Congress (rarely used), and the President influences the course of the legislation (a phenomenon that has been growing since the early 20th century) through powers like the State of the Union Address.
All the bureaucracy and different departments give the Executive access to more qualitative and quantitative information than any other branch.
Legislative Branch
The House and Senate, and their committees, formulate or pigeonhole legislation before bringing it to a debate on the floor and voting to pass or reject it.
Congress can override Presidential vetoes on legislation.
Congress can impeach the President (the House passes articles of impeachment, the Senate convicts or rejects them).
High reelection rates add to Congress members' seniority and gives them leverage to influence the President (who gets 8 years at most)
Only Congress can ratify treaties, which supersede Federal laws. But good luck convincing Native Americans of that.
Only Congress has the power to declare war. Congress has declared war 5 times among the hundreds of actions United States military forces have been part of.
Judicial Branch
Most importantly, the courts can strike down Federal and State legislation as unconstitutional through judicial review established in Marbury v Madison.
Theoretically judges can remain apolitical since their terms are for life. In some instances, like Eisenhower appointing Warren, a judge can completely defy the expectations of the President appointing him. But usually, Presidents know what they're getting based on a judge's previous court decisions and writings, and judges (like Anton Scalia) are as politically partisan as any politician.
That's basically what you need to know as a general overview. Specifically, in the case of the Iraq war, intelligence agencies like the CIA/FBI report to the Executive Branch, the Congress had access to much less information than the Executive, the Congress passed a resolution authorizing Bush to use force at his discretion, and that's what he did in March 2003.