Kodachrome was the brand name of a particular film technology released by Kodak in 1935 (after years of experimentation going back as far as the 1920s), and discontinued in 2009. The most popular and well-known format of this film was 35 mm slides, but it was used for other sorts of film, both still and motion-picture. Kodachrome was very popular in its time, and was memorialized in song. Alas, like Paul Simon's mom, Kodak did eventually take Kodachrome away. The film ended production in 2009, and on December 30, 2010, the last photo lab in the world with the capacity to develop Kodachrome ceased this service. The final roll was developed on 18 January, 2011. There are reportedly ways of developing Kodachrome in black-and-white chemistry, but as Simon sang, "Everything looks worse in black-and-white."
The developing process for Kodachrome was known as K-14, and attempts by "photo geeks" to bring a K-Lab machine back into service failed due to its highly specialized nature; it requires complex and hard-to-maintain equipment for which parts are no longer manufactured, and specialized chemicals that are no longer available. So you're out of luck if you ever want to try to develop it in its full-color form.
Scanning Kodachrome is the same as scanning any other slide film, with one caveat: do not use Digital ICE dust removal. Kodachrome's cyan layer absorbs the infrared light used in digital ICE just like dust does; digital ICE will attempt to interpolate these regions, giving weird results.