The United States Congress was originally intended by the Framers of the Constitution to be the most important of the three branches of government. During the 20th Century, however, the Executive and Judicial Branches steadily encroached upon the legislative "turf" of Capitol Hill. The relative power of Congress has waxed and waned in a cyclical fashion over the years, generally receding in times of war, when strong presidential authority is expected. The bicameral structure of Congress-- the House of Representatives vs. the Senate -- was a clear example of the desire by the Founding Fathers to restrain the exercise of government power, so as to preserve liberty. Those who express frustration at the slow pace of congressional work often forget this basic point. The U.S. government was not supposed to be "efficient."
Lawmaking: Congress may pass laws, subject to the limits set forth in the Constitution.
Budget: Both chambers have responsibilities for deciding on taxing and spending issues. The House plays a special role, inasmuch as all taxing bills must originate in the House.
Representation: Both chambers, and especially the House, serve to voice the concerns and wants of their constituents. If voters are pleased with what their legislators do in Washington, they are more likely to be reelected, and vice versa.
Advice and Consent: The U.S. Senate plays a special "Advice and Consent" role, having the power to confirm or reject presidential appointees for Cabinet posts (department secretaries, etc.), judgeships, ambassadors, and heads of other key Executive Branch agencies. The Senate also has the power to ratify or reject foreign treaties.
Oversight: Congressional committees can hold hearings to inquire into how Executive branch agencies are performing their duties.
Constituent service: Members of Congress try to help individual constituents get government services, cutting bureaucratic red tape, etc.
Leaders of each chamber for the subsequent two-year term are selected during the "lame-duck" session that follows each congressional election. In the Senate, the Majority Leader is in charge of floor proceedings, although there is greater flexibility with regard to procedure compared to the House. The Speaker of the House exercises strong control over the agenda. Both chambers delegate much of their work to various committees, but committees play an especially large role on the House side.
The Republicans took control of the House in the 2010 elections, and gained several seats in the Senate but fell short of a majority. An unusally sharp partisan divide, accompanied by increasing ideological polarization, has made it very difficult for Congress to reach agreement on the budget, debt ceiling, confirmation of presidential appointees, etc. The 2012 election left that situation unchanged, with a divided Congress and a Democratic president (Obama) who was reelected. The Democrats picked up two seats in the Senate.