The Smithsonian is located in Washington, D.C. by the National Mall a few blocks from the Capitol Building. It is one of several Smithsonian institutions located around the city. It served as a museum featuring exhibits on everything from natural history to the fine arts.
After the outbreak, survivors and refugees set up tents and makeshift shelters inside the museum, but they were quickly overrun and abandoned. Three and a half years into the apocalypse, survivors from the surrounding communities decided to go to the museum in order to procure ancient farming supplies and tools, such as a plow and a colonial wagon. The mission nearly goes awry when a glass floor breaks, nearly sending Ezekiel into a mass of walkers. However ultimately they are successful and leave the museum with their newly acquired tools.
While not explicitly named, interviews with the showrunners heavily imply that the museum is supposed to be the Smithsonian Museum of Natural History. It is strange then that the museum also includes exhibits dedicated to American history and fine arts, when in reality those are featured in separate museums.
The Georgia State Capitol was used for filming the interior of the museum, while the Atlanta Judicial Building was used for the exterior.