Washington DC is one of the most exciting cities in the world. It is also a top destination city for college students with every conceivable major. You may find yourself working at a think tank, a global association, a government agency or a congressman's office. Internships can be very competitive so plan ahead. Housing is typically available at local colleges and universities.
Your college or university may have a Washington internship program, so check that out first. Some schools have a combined academic and internship program. It might be called "Semester in Washington" or "ABC University in Washington". The various Washington programs within colleges are typically managed through either the Study Abroad Office, the Political Science Department, Internship Center or Career Services.
There are also many independent organizations that arrange internships in DC. Some of these are private companies and some are nonprofit organizations like the one listed below as an example.
Washington Internship Institute http://wiidc.org/admissions/
White House Internship- The most famous internship program http://www.whitehouse.gov/about/internships