Public Links are automatically created for any files put inside your Dropbox Public folder. Public links are read-only, and function essentially like most files on the internet – you get a direct link to download them, and that is all you can do with them
Shareable Links are manually created by you, and are essentially the same as Public Links (they’re read-only too), except that you can create links not only to files, but also to folders. You can see an example of a folder shared using shareable links here.
Shared Folders are a fantastic way to work together with people on files. Basically, everyone who is “in” the shared folder (has been invited to it and accepted the invitation) will have a special folder inside their Dropbox that is exactly the same for everyone—if you put a file inside that folder, everyone else gets it too; if someone else edits a file, everyone else gets the new edit. Shared folders are read and write for all members, and take up space in every user’s account.
There’s now a Dropbox Help Center article containing more information here.