The Public folder was a feature in Dropbox that was later removed. It now exists only for users who've signed up before it was disabled.
For newer accounts, you can share a link to any file from any directory by clicking Share Link. You can't, however, get a direct public link from the context menu itself.
See: What is the Public folder for? - Dropbox
Please note: New Dropbox accounts created after October 4, 2012 no longer have a Public folder.
If you want your link to be public, you don't have to do anything special. In the web page that pops up once you click Share Link, make sure you click Get Link to get the public link. You could also copy the URL that's displayed in your browser (the one starting with