This term applies equally to reader and audience member. Without the audience's willingness to believe in the fictional world he/she is about to experience either through pages or on stage, the story fails. The audience has to be open to the world--no matter how far-fetched or ridiculous it may appear. Think Star Wars and Lord of the Rings--if the audience wasn't willing to believe in galactic battles between good and evil or in hobbits and elves working with men against the evil of an all-powerful ring, neither of these fabulously popular stories would have taken off as they did.
The reader/viewer/listener must be willing to accept reality as the author has defined it, without questioning things that are unrealistic, supernatural, or otherwise outside the boundaries of true-life experience. For example, in Hamlet, the audience accepts the existence of the Ghost, even though we know that there are no such things. We accept the Ghost because it is part of the world of the play that we have willingly entered. We willingly suspend our disbelief in ghosts in order to participate as the audience of the play.
I think that the Coleridge's idea can go to the heart of what an author or artist feels is their perception of reality. When Coleridge developed the idea, he sought to link it to Romanticism and its element of imagination. Applied from this, one can see that the artist has an obligation to envision reality as it should be, as opposed to what it is. The "suspension of disbelief" is the duty of the artist to envision a world in any and all forms, and if it transcends reality, so be it. Audiences are not able to critique the work from "not real" points of view because that proves Coleridge's argument: Art is for its own sake, and if it redefines reality, so be it. In some sense, we can see many artists having adopted this idea and transformed the dynamic. Picasso was able to do this quite often and create strokes of genuis in the process.
I've always considered the "willing suspension of disbelief" in its literary application between writer and reader. If a reader is to enjoy, for instance, a gothic narrative, she must be willing to "suspend" disbelief in the supernatural. The attitude of "well, that can't happen" destroys any possibility of becoming engaged in the story. How could we enjoy a vampire story, for instance, if we couldn't for a while believe in vampires? In terms of the theater, a member of the audience must be willing to forget that she is watching a play and that the characters are only actors dressed up and walking around in a set. This temporary "suspension of disbelief" allows the audience to be transported from where they really are to another place and time and to become engaged in the story, whatever it happens to be.