Washington, D.C., as the nation's capital, has more monuments and statuary than you could see in a day, but do be sure to visit the Washington Monument (the world's tallest obelisk), the stately Lincoln Memorial, and the incredibly moving Vietnam Veterans Memorial. The city's architecture is also an attraction—the Capitol Building and the White House are two of the most iconic buildings in the country and often serve to represent the whole nation to the world.
Actually, a number of American cities have world-renowned skylines, perhaps none moreso than the concrete canyons of Manhattan, part of New York City. The site of the destroyed World Trade Center towers remains a gaping wound in Manhattan's vista, but the Empire State Building and the Chrysler Building still stand tall, as they have for almost a century. Chicago, where the skyscraper was invented, is home to the country's single tallest building, the (former) Sears Tower, and an awful lot of other really tall buildings. Other skylines worth seeing include San Francisco (with the Golden Gate Bridge), Seattle (including the Space Needle), Miami, and Pittsburgh.
Some human constructions transcend skyline, though, and become iconic symbols in their own right. The Gateway Arch in St. Louis, the Statue of Liberty in Manhattan, the Hollywood Sign in Los Angeles, and even the fountains of the Bellagio casino in Las Vegas all draw visitors to their respective cities. Even the incredible Mount Rushmore, located far from any major city, still attracts two million visitors each year.