There are many, but the African American History (free, but reserve in advance), Air & Space Museum, Natural History Museum, National Gallery of Art, Holocaust Museum, American Indian, and are a few of the more popular ones.
White House (view from outside), Washington Monument, MLK Monument, Lincoln Memorial, Jefferson Memorial, Vietnam War Memorial, Korean War Memorial, WWII Memorial, and Capitol Visitor's Center, National Mall, Tidal Basin.
DC is made up of many different neighborhoods with different history and culture. Some areas to explore include Georgetown, Dupont, Adam's Morgan, and U Street.