Permission is given by the entity that owns the original copyright, which may be the author, the publisher, or some third party that bought the rights to it. That could even be the person adapting it, or their publisher.
"Adapted" usually means updated and edited, perhaps with new examples or experiments, to meet a specific need (or to modernize it). For example, if technology has changed the field in question so a 1970 era textbook on sociology isn't good enough to teach a modern class, due to the advent of social media, the Internet, the acceptance of homosexuality and gay marriage, the changing sexual attitudes, etc. But enough of the textbook containing sociological methods, basic statistics and experiment design are still valid, so somebody just wants to "adapt" that textbook to modern times.