"Permission Culture" refers to the idea that people should ask permission before they use a copyrighted work. In fact, many, many uses of copyrighted works are perfectly legal without asking permission first, or at all.
Copyright grants the rightsholder a set of rights only over some actions (reproducing, distributing, preparing derivative works; and in some cases, performing, displaying, or digitally performing). But copyright law grants numerous exceptions and exemptions even from these rights -- distributing after the first sale (17 USC 109); preparing derivative works and reproducing in special formats for blind or print-disabled people (17 USC 121); reproducing and distributing for research, teaching, and library preservation (17 USC 108); and many, many more.
Chief among the the exceptions is "fair use" (17 USC 107), which states that "the fair use of a copyrighted work ... is not an infringement of copyright." Congress listed "purposes such as criticism, comment, news reporting, teaching (including multiple copies for classroom use), scholarship, or research" as examples of the kinds of uses that were fair, and then listed several criteria to consider in deciding if a use was fair. (See FAIR USE.)
If someone is not covered by an exception like fair use, then they can ask for permission (request a "license"), and may end up paying for permission.
Unfortunately, although there are many, many exceptions -- including fair use, which is quite broad -- people may be confused by the complexity of the law, or intimidated by the possibility -- even if remote -- of very large copyright damages. They may decide that it is easier or safer to ask permission. And sometimes it is, but often it is not. See, for example, this story about a musician who wanted to make a parody of a song, and decided to ask permission first. (Tim Cushing, July 30, 2012, "Dear Permission Culture: This Is Why No One Wants To Ask For Your OK", TechDirt.) In many countries -- including the US -- a parody is firmly protected as a fair use. Because a parody is a fair use, it's not copyright infringement, and therefore no permission is required to make a parody.
If people routinely ask for permission when none is needed, then a couple of things happen: One, a lot of unnecessary costs (money, time, hassle) are incurred in making uses that are actually okay. And two, a lot of uses won't happen that should, either because permission is denied or because it's just too hard to ask permission. These are significant downsides to the "permission culture". In fact, they are contrary to the very purposes of copyright law itself, which is intended to foster new works and new uses.
Last Edited: 3 February 2013