In 1846, James Smithson gave the American government a whole bunch of money. In return, he wanted some museums. Thus the Smithsonian Institution was born.
Most of the Smithsonian is on the mall, along with all the other cool but touristy stuff in Washington, D.C. (the Washington Monument, Capitol, Lincoln Memorial, etc.). All of the museums are free, and they are open 364 days per year -- closed only on Christmas. There are lots of cool musuems in the Smithsonian; if you're ever in DC, you owe it to yourself to check them out. After all, they're free.
The Big Five
The ones listed above are the cool parts of the Smithsonian. There are other, lesser, art museums, and other administrative buildings that have small exhibits. In no particular order...
... plus a couple of obscure ones. /msg me if you really care, and I'll add them. Most of the museums aren't noded yet. I'm working on it...
Finally, it's a museum, it's on the mall, and it's free -- but it's not part of the Smithsonian. It's the National Gallery of Art. And it's pretty cool, too.
See http://www.si.edu for all your Smithsonian needs | 
