How many days do I need to visit Washington DC?

If you want to hit the ground running and just get a brief view of DC, then you NEED at least two days. I outlined some sample itineraries/walking tours in this answer not long ago.
Basically, you need a half day to do the Monument/White House walk from the Washington Monument to the The White House, then back to the National World War II Memorial and Lincoln Memorial which is close to the Vietnam War Veteran's Memorial and Korean War Veterans Memorial then over to the Tidal Basin with its three monuments there. That relaxing walk if probably the most important DC walk. If your timing is right, you may be here for the Cherry Blossom Festival which brings crazy crowds, but the cherry blossoms in bloom around the Tidal Basin are just gorgeous.
In another half day, you can tour the United States Capitol and check out the Library of Congress' Jefferson building. It's my favorite building interior in DC, free, and totally worth stopping in to see it. You can easily take the tunnel from the Capitol to the Library too without even stepping outside. There are several decent restaurants behind the library on Pennsylvania Ave. I like We, The Pizza.
While you could easily spend an entire day in just ONE Smithsonian museum, if you're short on time, you can stop in 3-4 on the National Mall and get a taste of what they have. Don't miss the flag exhibit at National Museum of American History - it's right in front when you enter, will only take you 20-30 minutes to see and is amazing with the original flag that flew over Ft. Henry during the war. You may want to run upstairs to see the 1st Ladies dresses too. Run into the Museum of Natural History to see the Hope Diamond. Their gemology exhibit is huge, but my favorite. The National Museum of the American Indian has the best food on the mall and a really special museum as well. I'd check out the Smithsonian website ahead and see what's new and get a feel for what each museum has to offer, then prioritize and hit the ones you "need" to see.
If you do a half day at the mall, head over and spend a couple hours at Arlington National Cemetery. Walk to the Tomb of the Unknowns and see Robert F. Kennedy Gravesite and the Lee's House.
Those are the basics. IF you have more time (and you could easily fill a week+), then go hang out in Georgetown (Washington DC) and visit the Washington National Cathedral. The best paid museums are the Newseum (which has a huge piece of the Berlin wall and a radio tower from 9/11) and the International Spy Museum. An easy metro ride will get you to Alexandria (Virginia) which is also incredibly amazing, so if you have a third day, go there.