1. New edition
When a text is in the public domain, anyone--from you to the world's biggest publisher--can edit it and republish the edited version. When the edits are substantial enough, the edited work is deemed a "new edition", and gets a new copyright, dating from the time the new edition was created.
How substantial must the edits be to qualify as a "new edition"? That is for a court to decide in any particular case. Changing some punctuation or Americanizing British spelling would not qualify a work for a new edition. Theorizing something about Shakespeare and rewriting lots of lines in "Hamlet" to emphasize your point would make a new edition. In between those extremes is a grey area, where each new edition would have to be considered on a case-by-case basis.
A special case, that isn't quite a new edition, is when someone "marks up" a public domain text in, for example, HTML. Where this happens, the text is in the public domain, but the markup is copyrighted. We've already seen that when an editor adds footnotes to a public domain text, he owns copyright on the footnotes but not on the text: similarly, when he adds markup to the text, he owns copyright on the markup.
Translation is a common and justified special case of a new edition. When someone translates a public domain work from one language to another, they get a new copyright on the translation (but not on the original, of course, which stays in the public domain so that lots more people can use it.)