The type which is extremely common in popular culture today and is also very old is the borrowing of the content without the appropriation or quotation of the medium. The contemporary entertainment industry calls such borrowing “repurposing”: to take a “property” from one medium and reuse it in another.
A common example is the filmed versions of the Rowling’s books. There are no references to the books on which they are based because it would disrupt the continuity and the illusion of the immediacy.
Very interesting form of the remediation is the emergence of the musical movement Wizard Rock: wizard rock music.
Another type of the remediation is the representation of the older medium in digital form. The computer is offered a new means of gaining access to these older materials, as if the content of the older media could simply be poured into the new one. The digital medium wants to erase itself, so that the viewer stands in the same relationship to the content as he or she would if he or she were confronting the original medium.
Good example would be an e-book, a book published in electronic form.
New medium can remediate by trying to absorb the older medium entirely, so that the discontinuities between the two are minimized. But of course the new medium remains dependant on the older one. For example, the video games that are based on the Harry Potter movies.
Of course film is also trying to absorb and use digital technology: Harry Potter movies.
Remediation can occur within a single medium. For example, a film borrows from an earlier film or a writer borrows from another book (fanfiction).