Since 1800, this is where the legislative branch of American government (ie Congress) has met to write the country's laws. The lower House of Representatives (435 members) and upper Senate (100) meet respectively in the south and north wings of the building. Enter via the underground visitor center below the East Front Plaza. Guided tours of the building are free, but tickets are limited and there's often a long wait. It's best to reserve online in advance (there’s no fee).
The hour-long jaunt showcases the exhaustive background of a building that fairly sweats history. You'll watch a cheesy film first, then staff members lead you into the ornate halls and whispery chambers cluttered with the busts, statues and personal mementos of generations of Congress members.
To watch Congress in session, you need a separate gallery pass. US citizens must get one in advance or in person from their representative or senator; foreign visitors should take their passports to the House and Senate Appointment Desks on the upper level. Congressional committee hearings are more interesting (and substantive) if you care about what's being debated; check for a schedule, locations and to see if they're open to the public (they often are) at www.house.gov and www.senate.gov.
Security measures here are strict, including no food, liquid or bags larger than 18in by 4in.