The icon of New York (and one of its most often photographed skyscrapers) was not always known as the Flatiron Building. In 1902, when it was built for the Fuller Company, it was called the Fuller Building. The company only had its headquarters there till 1910 so the name eventually changed.
The area where the building is located (between the 22nd and 23rd Street) was already known as the "Flat Iron" and so the building got its name after the spot rather than after its shape. Nowadays though, the whole area is called the Flatiron District (after the building).
Sadly, it is not open to the public and there are no tours organised - the building is full of offices. Many publishing companies have their headquarters here (e.g. MacMillan Publishing). Still, it is definitely worth the trip because the Renaissance Revival building looks stunning even from the outside.